CAP 4932:  DESIGN OF BRAIN-LIKE SYSTEMS

 Oleg V. Favorov, Ph.D.

Fall 2002






GENERAL COURSE DESCRIPTION

Subject:  The structure and function of the brain as an information processing system, computation in neurobiological networks.

Objective:  To introduce students to basic principles of the mammalian brain design, information processing approaches and algorithms used by the central nervous system, and their implementation in biologically-inspired neural networks.

Schedule:   Tuesday, Thursday 14:30 - 15:45, Portable Classroom Bldg.10, room 104.

Prerequisites:  COP 3530 or instructor permission.  Good computer programming skills are required.

Teaching methods:   Lectures, development of computer models and their simulations.

Evaluation of student performance:  Homework (computer modeling projects, 50%), mid-term and final exams (50%), class participation.

Textbook:   Essentials of Neural Science and Behavior.  E. R. Kandel, J. H. Schwartz, T. M. Jessell; Simon and Schuster Co., 1995.
                  ISBN 0-8385-2245-9

Office hours:   Tuesday, Thursday 12:30 - 2:30 pm, room 242 Computer Science Building.

Contact:   phone 407-823-6495;  e-mail  favorov@cs.ucf.edu
 
 
 

COURSE CONTENT

    General layout of CNS as an information processing system
    Information processing in single neurons
        - Elements of membrane physiology
        - Compartmental modeling of dendritic trees
        - Synaptic input integration in dendritic trees
    Learning in neurobiological networks
        - Hebbian rule: associative learning networks
        - Error-Backpropagation learning
        - Self-organization of cortical networks
    Local network dynamics
        - Functional architecture of CNS networks (emphasis on cerebral cortex)
        - Nonlinear dynamical concepts in neural systems
        - Cerebral cortical dynamics
    Information processing in cortical output layers
        - SINBAD mechanism for discovery of causal environmental variables
        - Building an internal  model of the environment
        - Basic uses of an internal model
    Visual information processing
        - Stages and tasks of visual information processing
        - Distribution of roles among visual cortical areas
        - Functions and mechanisms of attention
    Memory
        - Classification of different forms of memory
        - Memory loci and mechanisms
    Motor control
        - Basics of muscle force generation and its neural control
        - Reflexes vs. internal models
        - Sensory-motor integration in the cerebral cortex
        - Limbic system and purposeful behaviors


Lecture 1, August 20, 2002:   NEURON

Neuron’s function – to receive information from some neurons, process it, and send it to other neurons

Subdivisions of a neuron’ body:
· Dendrites – receive and process information from other neurons
· Soma (cell body) – combines all information
· Axon hillock – generates output signal (pulse)
· Axon – carries output signal to other neurons
· Axon terminals – endings of axonal branches
· Synapse – site of contact of two neurons

Neurons communicate by electrical pulses, called ACTION POTENTIALS (APs) or SPIKES.
Spikes are COMMUNICATION SIGNALS.
All spikes generated by a neuron have the same waveform; information is encoded in timing and frequency of spike discharges from axon hillock.

NEURON ELECTROPHYSIOLOGY

Neurons are electrical devices.
Inside of a neuron (cytoplasm) is separated from outside extracellular fluid by neuronal membrane.
Neuronal membrane contains numerous ion pumps and ion channels.
Na+/K+ ion pump – moves Na+ ions from inside to outside and K+ ions from outside to inside a neuron.  As a result, these pumps set up Na+ and K+ concentration gradients across the membrane.
Ion channels – pores in the membrane that let ions move passively (by diffusion) across the membrane.  Each type of channel is selectively permeable to one or two types of ions only.
Ion channels can be open or closed, (“gated”).  In different types of channels the gate can be controlled by voltage across the membrane or by special chemical compounds or mechanically (in sensory receptors).  Some types of ion channels are always open and are called “restingchannels.

RESTING ION CHANNELS
Neurons have K+  Na+  and  Cl-  resting ion channels.
Concentration gradients of Na+ and K+ across the neuronal membrane drive ions across the membrane through ion channels and set up an electrical gradient, or a MEMBRANE POTENTIAL (V).
The value of the membrane potential is determined by permeability of resting ion channels and typically is around –70 mV.  It is called RESTING POTENTIAL (Vr).
        Vr = -70 mV  - it is “functional zero” or baseline.

Text: pp.21-40 (overview)
For an in-depth reading (optional):115-116 (ion channels), 133-134, 136-139 (membrane potential)


Lecture 2, August 22, 2002:   MEMBRANE ELECTROPHYSIOLOGY

GATED ION CHANNELS

Opening gated channels lets ions to flow through them and results in a change of the membrane potential.

        HYPERPOLARIZATION – making membrane potential more negative.
        DEPOLARIZATION – making membrane potential less negative.

EQUILIBRIUM POTENTIAL (E) of a particular class of channels is such a membrane potential V at which there is no net ion flow across open channels.
        ENa+ , equilibrium potential of Na+ channels is +40 mV.
        EK+ , equilibrium potential of K+ channels is -90 mV.
        ECl-, equilibrium potential of Cl- channels is -70 mV.

INTERNAL SIGNALS used by neurons are carried by membrane potential V.  The signal is a deviation of membrane potential from its resting level Vr =  –70 mV.
        - positive deviation is depolarization, or excitation
        - negative deviation is hyperpolarization, or inhibition

Mechanism of generating internal signals is by opening ion channels.  Each type of channel is selectively permeable to certain ions.  Each ion has a specific equilibrium potential E.  When a particular type of channels is opened, V will move towards E of those channels.
        To raise V:   open Na+ channels.
        To lower V:   open K+ channels.
        To scale down  (V – Vr) :   open Cl- channels.

MECHANISM OF ACTION POTENTIAL GENERATION

APs are produced by 2 types of channels:
        1.   Voltage-gated Na+ channels.  Open when V is above –55 mV.  Open very quickly, but only transiently (stay open only about 1 msec).  Need resetting by lowering V below –55 mV.
        2.   Voltage-gated K+ channels.   Open when V is above –55 mV.  Slower to open, but not transient.  Do not need resetting.

AP is triggered when V exceeds –55 mV.   Thus, –55 mV is the AP THRESHOLD.
APs are ALL-OR-NONE – i.e., have uniform shape.
ABSOLUTE REFRACTORY PERIOD (about 2 msec) – period after firing a spike during which it is impossible to trigger another spike.
RELATIVE REFRACTORY PERIOD (about 15 msec) – period after firing a spike during which AP threshold is raised above –55 mV.
Frequencies of firing APs:   up to 400 Hz, but normally less than 100 Hz.

SYNAPTIC TRANSMISSION

Brains of mammals, and cerebral cortex in particular, mostly rely on chemical synapses.
Source neuron is called PRESYNAPTIC NEURON and target neuron is called POSTSYNAPTIC NEURON
Presynaptic axon terminal has synaptic vesicles filled with chemical compound called TRANSMITTER.  Presynaptic membrane is separated from postsynaptic membrane by synaptic cleft.  Postsynaptic membrane has RECEPTORS and RECEPTOR-GATED ION CHANNELS.
When an AP arrives to the presynaptic terminal, it makes a few of the synaptic vesicles to release their transmitter into the synaptic cleft. In the cleft, transmitter molecules bind with receptors and make them to open ion channels.  Open ion channels let ions flow through them, which changes membrane potential V.
AP-evoked change of V is called POSTSYNAPTIC POTENTIAL (PSP).
Positive change of V is called EXCITATORY PSP (EPSP).
Negative change of V is called INHIBITORY PSP (IPSP).
Transmitter stays in the synaptic cleft only a few msec and then escapes or is taken back into the presynaptic terminal and is recycled.

Text:   pp. 31-39 (overview).
If want extra information, see  pp. 115-116 (ion channels), pp. 133-134, 135-139 (membrane potential), pp. 168-169 (action potential), pp. 181-182, 191 (synapse), pp. 227-234 (synaptic receptors and channels).



Lecture 3, August 27, 2002:   MEMBRANE ELECTROPHYSIOLOGY

SYNAPTIC INTEGRATION

A postsynaptic potential (PSP), evoked by an action potential at a given synapse, spreads passively (electrotonicly) throughout the neuron’s dendrites and eventually reaches the axon hillock, where it contributes to generation of action potential.
Because of its passive spread, PSP fades with distance from the site of its origin, so its size is reduced significantly by the time it reaches the axon hillock.
A single synapse generates only very small PSPs (typically, less than 1 mV).  However, any given neuron receives thousands of synaptic connections and together they can add up their PSPs and depolarize axon hillock sufficiently to trigger action potentials.
SPATIAL SUMMATION – addition of PSPs occurring simultaneously at different synapses.
TEMPORAL SUMMATION – temporal buildup of PSPs occurring in rapid succession.

SYNAPTIC EFFICACY

Synaptic efficacy = connection “weight” (or strength) – how effective is a given synapse at changing membrane potential at the axon hillock.
Major determining factors:
 1.   amount of transmitter released from the presynaptic terminal by one spike.
 2.   number of postsynaptic receptors/channels
 3.   distance to the axon hillock

BRAIN DESIGN PRINCIPLES:

- A synapse transmits information in one direction only.
- A given synapse can only be excitatory or inhibitory, but not both.
- A given neuron can only make excitatory or inhibitory connections, but not excitatory on some cells, inhibitory on others.
- Synapses vary in their efficacy, or weights.



Lectures 4-5, August 29-September 3, 2002: NEURON AS AN ELECTRICAL CIRCUIT

An ion channel can be represented as an electrical resistor of a particular conductance, connected in series with a battery, whose charge is equal to the equilibrium potential E of this ion
channel.
Different ion channels in the same membrane can be represented by resistors/batteries connected in parallel to each other.
All the resting channels (K+  Na+  Cl-) can be represented together by a single, lumped resistor of conductance gm, in series with battery whose E = -70 mV, or resting membrane potential.
gm is called “passive membrane conductance”
A synapse can be represented by a resistor, whose conductance is equivalent to that synapse’s weight (efficacy), in series with a battery and a switch, controlled by presynaptic action
potentials.
All the synapses that use the same receptors and ion channels can be represented together by a single, lumped variable resistor of conductance G, in series with a battery.
  G = S wi * Ai,   where wi (or gi) is the weight of synapse i and Ai is the activity of the presynaptic cell i.

A neuron can be represented by an electrical circuit that consists of a number of components all connected in parallel:
- capacitor Cm, representing membrane capacitance
- resistor gm in series with a battery Er = -70 mV, representing passive membrane resistance
- variable resistor Gex in series with a battery Eex = 0 mV, representing net conductance of all the excitatory synapses
- variable resistor GinK in series with a battery EinK = -90 mV, representing net conductance of all the inhibitory synapses that use “subtracting” K+ ion channels
- variable resistor GinCl in series with a battery EinCl = -70 mV, representing net conductance of all the inhibitory synapses that use “dividing” Cl- ion channels
- variable resistor gAPNa in series with a battery ENa = +40 mV, representing voltage-gated Na+ channels responsible for generation of action potential
- variable resistor gAPK in series with a batteryEK = -90 mV, representing voltage-gated K+ channels responsible for generation of action potential
- other components, if known and desired to be included in such a model

This circuit representation of a neuron is called POINT NEURON MODEL, because it treats a neuron as a point in space, dimensionless, i.e., it ignores the dendritic structure of neurons.

If for simplicity (because of its minor effect) we ignore contribution of membrane capacitance, then we can compute membrane potential V that is generated by this circuit as:

                    gm* Er + Gex * Eex + GinK* EinK + GinCl* EinCl + gAPNa* ENa + gAPK* EK
        V  =     --------------------------------------------------------------------------------------
                              gm + Gex + GinK + GinCl+ gAPNa + gAPK

Text:  OPTIONAL  pp. 142 – 147 (representing resting channels),  pp. 213 – 216 (representing synapses; discussed on an example of synapses between neurons and muscles, called end-plates; they use
transmitter called acetylcholine, or ACh).



Lecture 6, September 9, 2002:   MODELING PROJECT #1 - POINT NEURON MODEL

Write a computer program to simulate a neuron modeled as point electric circuit.

Parameters:      number of excitatory input cells - 100
                        number of inhibitory input cells - 100
                        time constant, t - 4 msec
                        gm = 1
                        assign connection weights w+ and w- of excitatory and inhibitory input cells randomly in the range between 0 and 1
                        set Gex = 0 and Gin = 0 before the start of simulation

Simulation:        simulate time-course of membrane potential with time step of 1 msec
                        At each time step, do the following 3 procedures -
                        1.  Pick randomly which of the input cells have an action potential at this point in time.
                            For these cells, set their activity A = 1; for all the other cells, set their activity A = 0
                            Active cells should be chosen such that approximately 10% of them will have an action potential at any given time.

                        2.  Update Gex and Gin  :
                                    Gext = (1-1/t) * Gext-1  +  (1/t) * Cex * S(w+ *  A+i t )

                                    Gint = (1-1/t) * Gint-1  +  (1/t) * Cin * S(w-  *  A-i t )

                        3.  Update deviation of membrane potential from resting level, DV :

DV = ( 70 * Gext ) / ( Gext +  Gint  + gm )

Exercises:
           1.  Testing the program.
                                        Set all w+'s and w-'s to 1 (i.e., all connections should have the same weight, =1).
                                        Set Cex = 1   and  Cin = 2
                                        Run the program 20 time steps and plot DV as a function of time (Plot #1).
                                        Hint:  if the program is correct, DV should raise quickly from 0 to around 22-23 mV and stay there.

            2.  Effect of  Cex.
                                        Randomize all w+ 's and  w- 's in the range between 0 and 1.
                                        Set Cin = 0  (i.e., inhibition is turned off).
                                        Run the program with many (e.g., 20) different values of Cex.
                                        In each run, do 20 time steps and save the value of DV  on the last, 20th time step (by thenDV should reach a steady state).
                                        Plot this value of DV as a function of Cex on a log scale. You should find such a range of Cex in which DV will start at approximately 0 and
                                        then at some point will raise sigmoidally to 70 mV. This is Plot #2.

            3.  Effect of Cin.
                                        Same as Exercise #2, but set Cex to such a value at which DV in  Exercise #2 was approximately 60 mV.
                                        Run the program with many different values of Cin.
                                        Plot DV as a function of Cin on a log scale. You should find such a range of Cin in which DV will start at approximately 60 and
                                        then at some point will descend sigmoidally to 0 mV. This is Plot #3.

Submit for grading:         brief summary of the model
                                        description of the 3 exercises and their results (plots #1, 2, 3)
                                         text of the program
Due date:   September 19.
 

                                             PROGRAM PSEUDO-CODE (if you want it)

Define arrays:       Aex(1-100)    - activities of excitatory input cells
                            Ain(1-100)      - activities of inhibitory input cells
                            Wex(1-100)   - excitatory connection weights
                            Win(1-100)    - inhibitory connection weights

Set parameters:    Nex = 100        - number of excitatory input cells
                            Nin = 100        - number of inhibitory input cells
                            TAU = 4          - time constant
                            Ntimes = 20   - number of time steps
                            Cex = 1           - excitatory scaling constant (in Exercise 2, vary the value systematically)
                            Cin = 2            - inhibitory scaling constant (in Exercise 2, Cin = 0; in Exercise 3, vary systematically)
                            Gm = 1           - passive membrane conductance

Assign random weights to excitatory and inhibitory connections:
        FOR I = 1 … Nex
                    Get random number RN in range [0 … 1]
                    Wex(I) = RN     (in Exercise 1, Wex(I) = 1)
        NEXT  I

        FOR I = 1 … Nin
                    Get random number RN in range [0 … 1]
                    Win(I) = RN     (in Exercise 1, Win(I) = 1)
         NEXT  I

Initialize excitatory and inhibitory conductances:
        Gex = 0
        Gin = 0

Compute DV for Ntimes  time steps:

         FOR Itime = 1 … Ntimes

            Choose activities of input cells at random with probability of 10% of A = 1:
                            FOR I = 1 … Nex
                                        Get random number RN in range [0 … 1]
                                        Aex(I) = 1
                                        IF ( RN > 0.1 ) Aex(I) = 0
                            NEXT I

                            FOR I = 1 … Nin
                                        Get random number RN in range [0 … 1]
                                        Ain(I) = 1
                                        IF ( RN > 0.1 ) Ain(I) = 0
                            NEXT I

            Sum all excitatory inputs:
                            SUM = 0
                            FOR I = 1 … Nex
                                       SUM = SUM + Wex(I) * Aex(I)
                            NEXT I

            Update excitatory conductance:
                            Gex = (1 – 1/TAU) * Gex + (1/TAU) * Cex *SUM

            Sum all inhibitory inputs:
                            SUM = 0
                            FOR I = 1 … Nin
                                        SUM = SUM + Win(I) * Ain(I)
                            NEXT I

            Update inhibitory conductance:
                            Gin = (1 – 1/TAU) * Gin + (1/TAU) * Cin *SUM

            Calculate DV:
DV = (70 * Gex)/(Gex + Gin + Gm)

             In Exercise 1, plot DV as a function of time step (Plot #1)

        NEXT Itime

In Exercises 2 and 3, plot final DV as a function of Cex (Plot #2) or Cin (Plot #3).



Lecture 7, September 12, 2002:   NEURON’S COMPLEXITIES
 

MULTIPLICITY OF ION CHANNEL TYPES

A neuron has large number (probably on the order of 100) of different types of ion channels.  These channels differ in:
- kinetics (faster-slower)
- direction of membrane potential change when open (towards its equilibrium potential)
- factors that control channel opening (transmitter, V, etc.)

2 most common ion channels in excitatory synapses of the cerebral cortex:
        channel type:      AMPA              NMDA
        transmitter:         glutamate           glutamate
        receptor:            AMPA              NMDA
        ions:                  Na (+K)             Na  Ca  (+K)
        kinetics:             fast                     approx. 20 times slower
        controls:            transmitter           transmitter, membrane potential (needs depolarization to open)

How would we modify temporal behavior equations in our program to incorporate these 2 channel types:

                                    GAMPAt = (1-1/tAMPA) * GAMPAt-1  +  (1/tAMPA) * CAMPAS(w+i  *  A+i t )

                                    GNMDAt = (1-1/tNMDA) * GNMDAt-1  +  (1/tNMDA) * CNMDAS[(w+i  *  A+i t ) * DV]+
 

ADAPTATION

Neurons respond transiently to their inputs and gradually reduce their firing rate even when the input drive is constant
Mechanisms:         - presynaptic adaptation
                            - postsynaptic receptor/channel desensitization
                            - AFTERHYPERPOLARIZATION (AHP)

Afterhyperpolarization is produced by several types of ion channels. They all are hyperpolarizing K+ channels, but have
different kinetics (fast – 15 msec, medium – 100 msec, slow – 1 sec, very slow >3 sec).
They all are opened by membrane depolarization (especially by action potentials).

Text:  pp. 149 – 159 (this is optional, if you do not understand something and want to clarify it).



Lecture 8, September 17, 2002:   COMPARTMENTAL MODELING OF A NEURON

DENDRITIC TREES

How to model dendritic trees?  A branch of a dendrite is essentially a tube made up of insulator – membrane.
To think about it as an electrical circuit, we can subdivide it into small compartments.
These compartments become essentially dimensionless and each can be represented by the point model.
Then a dendrite can be represented as a chain of point electrical circuits connected with each other in series via longitudinal conductances.
The entire neuron then is a branching tree of such chained compartments.  Such a model of a neuron is called a COMPARTMENTAL MODEL.

NEURON'S INTEGRATIVE FUNCTION

Neuron is a set of several extensively branching trees of electrical compartments, all converging on a single output compartment.
Each compartment integrates its synaptic inputs and inputs from the adjoining compartments.
This integrative function is complex, nonlinear, and bound between –90 and +40 mV.
Thus, membrane potential of a compartment i, Vi = f (Vof adjoining compartments, synaptic inputs, local ion channels, etc.)
Bottom line:  dendritic trees, and neuron as a whole, can implement a wide range of complex nonlinear integrating functions over their synaptic inputs.

Text:  pp. 149 – 159 (this is optional, if you do not understand something and want to clarify it).
 
 

HOMEWORK ASSIGNMENT #2

  2 most common ion channels in inhibitory synapses of the cerebral cortex:

        channel type:      GABAA             GABAB
        transmitter:         GABA                GABA
        receptor:             GABAA            GABAB
        ions:                   Cl                       K
        kinetics:              fast                     approx. 40 times slower
        controls:             transmitter           transmitter

Draw a connectional and electrical circuit diagrams of a POINT neuron with 3 sets of connections:
(1)  excitatory (AMPA and NMDA channels)
(2)  inhibitory (GABAA and GABAB channels)
(3)  inhibitory (GABAA channels only)

Also include in the electrical diagram:
(1)  action potential-generating channels
(2)  medium afterhyperpolarization channels for adaptation
(3)  slow afterhyperpolarization channels for adaptation

Write all the equations necessary to describe this neuron (ignore membrane capacitance for simplicity).

Due date:  September 26
 
 

HOMEWORK ASSIGNMENT #3

COMPARMENTAL MODELING OF A PYRAMIDAL CELL

The task is to draw an electrical circuit diagram representing a hypothetical PYRAMIDAL CELL from the cerebral cortex.
Pyramidal cells are the principal class of cells in the cerebral cortex (80% of all the cells there).
They have the body in a shape of a pyramid, out of whose base grow 4-6 BASAL DENDRITES and out of whose apex grows APICAL DENDRITE.
The apical dendrite is very long; it grows all the way to the cortical surface, near which the apical dendrite sprouts a clump of dendrites called TERMINAL TUFT.
The particular cell that we want to model is such a cell, but with only one basal dendrite (to reduce amount of work for you drawing all these dendrites).

The assignment is to model this cell with 4 electrical compartments:
            compartment S (representing soma) is connected with compartment B (representing the basal dendrite)
            and with compartment A (representing apical dendrite).
            Compartment A in turn is connected with compartment T (representing terminal tuft).

This neuron has following components in its compartments:
          Excitatory connections in:       T     A     B
          Inhibitory K+ connections in:  T     A     B
          Inhibitory Cl- connections in:  T     A     B     S
          Action potential channels in:                          S
 

Due date:  October 3.  Submit a drawing of the electrical circuit representing this cell. No comments are required.



Lecture 9-12, September 19-30, 2002:   FUNCTIONAL SUBDIVISIONS OF CNS

Divisions of the Nervous System:

           NERVOUS SYSTEM consists of (1) PERIPHERAL NERVOUS SYSTEM and
                                                                (2) CENTRAL NERVOUS SYSTEM (CNS)

            Peripheral NS consists of (1) SENSORY NERVES,
                                                    (2) MOTOR NERVES to skeletal muscles, and
                                                    (3) GANGLIA and NERVES OF AUTONOMIC NERVOUS SYSTEM (it controls internal organs)

            Central Nervous System has 2 subdivisions:  BRAIN and SPINAL CORD

            Brain has following major subdivisions:  BRAINSTEM, CEREBELLUM, DIENCEPHALON, two CEREBRAL HEMISPHERES
 

Sensory Systems

There are 6 sensory systems: SOMATOSENSORY (touch, body posture, muscle sense, pain, temperature), VISUAL, AUDITORY,
OLFACTORY, GUSTATORY, VESTIBULAR
 

CEREBRAL CORTEX

Cerebral cortex is a thin (approx. 2mm thick), but large in area layer of neurons.
In more advanced mammals (like cats, apes, humans) this layer is convoluted (to pack more surface into the same volume).
The folds are called sulci (singular is sulcus), the ridges are called gyri (singular is gyrus).
Major partitions of the cerebral cortex are called LOBES. There are 6 lobes:
         FRONTAL LOBE (planning and motor control)
         PARIETAL LOBE (somatosensory + high cognitive functions)
         OCCIPITAL LOBE (visual + high cognitive functions)
         TEMPORAL LOBE (auditory + high cognitive functions)
         INSULA (polysensory)
         LIMBIC LOBE (emotions)
Cortex is further subdivided into smaller regions, called CORTICAL AREAS.  There are about 50 of these areas.
Cortical areas are divided into:
         PRIMARY CORTICAL AREAS (handle initial sensory input or final motor output)
         SECONDARY CORTICAL AREAS (process output of primary sensory areas or control primary motor area)
         ASSOCIATIVE CORTICAL AREAS (all the other areas)
 

SENSORY PATHWAYS

SENSORY RECEPTORS -->  PRIMARY AFFERENT NEURONS -->  SENSORY NUCLEUS in spinal cord or brainstem -->
 --> RELAY NUCLEUS in THALAMUS -->  PRIMARY SENSORY CORTICAL AREA -->
 --> SECONDARY SENSORY CORTICAL AREAS -->ASSOCIATIVE CORTEX
 

MOTOR CONTROL SYSTEM

Motor control system contains 3 major subsystems:

1)   PREFRONTAL CORTEX -->  PREMOTOR CORTEX  -->  PRIMARY MOTOR CORTICAL AREA  -->
--> MOTOR NUCLEI in brainstem (for head control) or in spinal cord (for the rest of the body control)  -->  SKELETAL MUSCLES

             Motor nucleus – contracts one muscle
             Primary motor cortex (MI) – control of single muscles or groups of muscles
             Premotor cortex – spatiotemporal patterns of muscle contractions (e.g., finger tapping)
             Prefrontal cortex – behavioral patterns, planning, problem solving
 

2)   Entire cortex  -->  BASAL GANGLIA  -->  VA nucleus in thalamus  --> prefrontal cortex
                                                                     -->  VL nucleus in thalamus  -->  premotor and motor cortex

            Function of basal ganglia – learned behavioral programs, routines, habits (e.g., writing, dressing up)
 

3)   Somatosensory information from receptors -->
                                                                                 CEREBELLUM
      Premotor, motor, and somatosensory cortex  -->
 

                                  --> VL nucleus in thalamus  -->  premotor and motor cortex
     CEREBELLUM
                                 -->  motor nuclei

            Functions of cerebellum – learned motor skills
                                - movement planning
                                - muscle coordination (e.g., not to loose balance while extending arm)
                                - comparator function (compensation of errors during movements)
 

MOTIVATIONAL SYSTEM

            HYPOTHALAMUS in diencephalon – monitors and controls body’s needs (food, water, temperature, etc.)
           AMYGDALA and LIMBIC CORTEX in cerebral hemispheres – emotions, interests, learned desires
           RETICULAR FORMATION in brainstem – arousal (sleep-awake), orienting reflex, focused attention

Text:  pp. 10 –11, 77 - 88



Lectures 13-15, October 3-10:    CORTICAL TOPOGRAPHY

The subject of topography is how sensory inputs are distributed in the cortex.
The 2 basic principles are:
(1)   different sensory modalities (e. g., vision, auditory, etc.) are first processed separately in different parts of the cortex, and only after this processing they are brought together in higher-level
associative cortical areas.
(2)   within each sensory modality, information from neighboring (and therefore more closely related) sensory receptors is delivered to local cortical sites within the primary sensory cortical area,
and different cortical sites process information from different groups of sensory receptors.  The idea here is at first to bring together only local sensory information, so that cortical cells can
extract local sensory features (e.g., local edges, textures). In the next cortical area, cells then can use these local features to recognize larger things (e.g., shapes, objects), and so on.

So, projections from sensory periphery (skin, retina, cochlea) to primary sensory cortical are topographically arranged; however, they are not POINT-TO-POINT but SMALL
AREA-TO-SMALL AREA.  Next, from one cortical area to the next, still it would not be useful to mix widely different (unrelated) inputs. So, projections from lower to higher cortical areas are
also topographic.

As a result of such a distribution of inputs to cortical areas, these areas have TOPOGRAPHIC MAPS:
                in somatosensory cortex – SOMATOTOPIC MAPS
                in visual cortex – RETINOTOPIC MAPS
                in auditory cortex – TONOTOPIC MAPS.

RECEPTIVE FIELD (RF) of a neuron (or a sensory receptor) is the area of RECEPTOR SURFACE (e.g., skin, retina, cochlea) whose stimulation can evoke a response in that neuron.
Sensory receptors have very small RFs. Cortical cells build their RFs from RFs of their input cells; as a result, RFs of neurons in higher cortical areas become larger and larger.

RECEPTIVE FIELD PROFILE – distribution of the neuron’s responsivity across its receptive field (i.e., how much it responds to stimulation of different loci in its RF).

TOPOGRAPHY

In generating topographic maps in target cortical areas, the sensory axons are guided to the right cortical area and to the general region in that area by chemical clues.
This is the basic mechanism for interconnecting different parts of the brain and for laying the topographic map in each area in a particular orientation (e.g., in the primary somatosensory cortex,
the topographic map in all individuals is oriented the same way, with foot most medial, head most lateral in the area).

The basic mechanism for finer precision in topographic maps is by axons that originate from neighboring neurons (or sensory receptors) traveling together and then terminating near to each other.

This way, original neighborhood relations in the source area can be easily preserved in their projections to the target area.

In addition to this genetic mechanism of generating topographic maps in cortical areas (as well as in all the other parts of the CNS), there is also a learning mechanism.
This mechanism can adjust the topographic map to the particular circumstances of an individual.
For example, if a person lost an arm, the regions of somatosensory cortex that process information from that arm should be re-wired to receive information from other body regions.
Or, if a person preferentially makes use of some part of the body (e.g., a pianist's use of fingers), it would be desirable to send information from that part to a larger cortical region for improved
processing.

The mechanism for such individual tuning of topographic maps is SYNAPTIC PLASTICITY (change of connections in response to experience).
Connections can be changed by:
                        (1) axon sprouting (growing new axon branches, selective elimination of some other existing branches)
                        (2) changing efficacy of the existing synapses.

Text:  pp. 86-87, 99-104, 324-329, 369-375.



               MODELING PROJECT #2: RECEPTIVE FIELD OF A NEURON

                           HOMEWORK ASSIGNMENT #4

Write a computer program to compute a RECEPTIVE FIELD PROFILE of a target neuron that receives input connections from a set of excitatory source neurons.
Model parameters:     N = 40  - number of source cells.
                                RFR = 3  - receptive field radius.
D = 1  - spacing between receptive fields of neighboring neurons.

Weights of connections of source cells to the target cell, w(i), should be chosen randomly and then normalized so that Sw(i) = 1

Activity of source cell i is:  As(i) = 1 - (|S - RFC(i)| / RFR)
              where S - stimulus location on the receptor surface,
                        RFC(i) - receptive field of cell i.  RFC(i) is computed as RFC(i) = RFR + (i – 1) * D
           As is “instantaneous firing rate (or frequency)”; it is calculated as an inverse of time interval between successive action potentials.
                                If  As(i) < 0,  set  As(i) = 0.

The target cell is modeled as a point electrical circuit, made up of passive membrane conductance gm = 1 (E = -70) and net excitatory conductance Gex (E = 0).
Activity of the target cell is: DV = 70 * Gex / (Gex + gm)

In the steady state, Gex would be Gex = Cex * S w(i) As(i),  where w(i) is the connection weight of source cell i on the target cell.
To reflect temporal behaviors of ion channels,
                             Gex = (1 – 1/t) * Gex + (1/t) * Cex * S w(i) As(i)
                                        Use t = 4.

The task is to map the receptive field profile of the target cell.

Program flow:
Phase 1:    Set all the parameters (RFR, RFC(i), w(i), D, N, t, Cex)
Phase 2:   Deliver 450 point stimuli, spread evenly across the entire receptor surface.
                            The entire length of the receptor surface is RFR + (N – 1) * D + RFR = 45.
                            Therefore, space stimulus locations 0.1 distance apart:  S1=0.1, S2=0.2 … S450=45.
                For each stimulus location:
                              Compute activities of all the source cells, As(i)
                              Set Gex = 0
                              Do 20 time steps, updating Gex and DV.
                              Plot DV after 20th time update as a function of stimulus location S.

Submit for grading:
        description of the model
        plot of DV  as a function of S (choose Cex such that DV is in approx. 30 mV range)
        text of the program
               ........................................................................................................
                                               PROGRAM PSEUDO-CODE
                                                MODELING PROJECT #2
 

Define arrays:        As(1-40)     -  activity of sorce cell i.
                            W(1-40)   -  connection weight from sorce cell i to the target cell.
                            RFC(1-40)     -  receptive field center of source cell i.

Set parameters:    N = 40                  - number of source cells
                            Nstimuli = 450      - number of stimuli
                            TAU = 4            - time constant
                            Gm = 1              - passive membrane conductance
                            Cex = ?             - excitatory scaling constant
                            RFR = 3                 - receptive field radius
                            D = 1                     - spacing of receptive fields
                            RSL = 2 RFR + (N - 1) D    - receptor surface length

Assign random weights to all connections.
                                        SUM = 0

                                        FOR I = 1 ... N
                                                Get random number RN in range [0 … 1]
                                                SUM = SUM + RN
                                                W(I) = RN
                                        NEXT I

                                        FOR I = 1 ... N
                                                W(I) = W(I) / SUM
                                        NEXT I

Assign receptive field centers to source cells:
                                        FOR I = 1 ... N
                                                RFC(I) = RFR + (I-1) * D
                                        NEXT I

Present 450 stimuli and compute DV of the target cell for each stimulus:

         FOR Istimulus = 1 … Nstimuli

            Choose stimulus location:
                                        S = Istimulus*0.1

            Compute activities of source cells:
                                       FOR I = 1 … N
                                                As(I) = 1 - (|S - RFC(I)| / RFR)
                                                IF (As(I) < 0) As(I) = 0
                                        NEXT I

            Compute membrane depolarization DV of the target cell for 20 time steps:
                                        use the program you developed in the first project

            Plot DV (taken after 20 time steps) as a function of S

        NEXT Istimulus

END



OCTOBER 22:  MIDTERM EXAM
 

Sample questions:

Which of the following is (are) correct?   (e)
a) The function of a neuron’s dendrites is to transmit information to other neurons;
b) Action potentials normally are triggered in neurons’ dendrites;
c) Information, transmitted by a neuron, is encoded in the shape of its action potentials;
d) All of the above are correct;
e) None of the above is correct.

Which of the following is (are) correct?   (c)
a) Ion channels are pores in the cell membrane, and as a result they let all types of ions to pass through them;
b) Resting ion channels can be opened by a change in the membrane potential;
c) Membrane potential is the carrier of internal signals processed by dendrites and soma of a neuron;
d) All of the above are correct;
e) None of the above is correct.

Which of the following is (are) correct?   (a)
a) Changes in membrane potential are effected by opening and/or closing of gated ion channels;
b) Opening of an ion channel results in a change of the membrane potential away from the equilibrium potential of that channel;
c) Action potential is triggered when the membrane potential rises above the threshold of –35 mV;
d) All of the above are correct;
e) None of the above is correct.

Which of the following is (are) correct?   (e)
a) At the site of a synapse, membranes of the presynaptic and postsynaptic cells fuse together into a single membrane;
b) Transmitter molecules, released from the presynaptic terminal by an action potential, are absorbed by the postsynaptic cell;
c) A convergence at the axon hillock of postsynaptic potentials evoked simultaneously at several different synapses is called temporal summation;
d) All of the above are correct;
e) None of the above is correct.

Which of the following is (are) correct?   (a)
a) An efficacy of a synapse is determined, among other factors, by the numbers of receptors and receptor-gated ion channels at the site of that synapse in the postsynaptic membrane;
b) All the synapses made by the same neuron on other neurons have the same synaptic efficacy;
c) A neuron performs an essentially linear summation of the postsynaptic inputs it receives from other neurons; it is, fundamentally, a linear summator;
d) All of the above are correct;
e) None of the above is correct.

Which of the following is (are) correct?   (b)
a) Thalamus is a subdivision of the cerebral hemispheres;
b) Limbic cortex is an example of associative cortical areas;
c) Basal ganglia send their output directly to the cortex;
d) All of the above are correct;
e) None of the above is correct.

Which of the following is (are) correct?   (c)
a) Somatosensory cortical areas contain tonotopic maps of body;
b) Sensory receptors do not have receptive fields;
c) Amygdala and limbic cortex are involved in generation of emotions;
d) All of the above are correct;
e) None of the above is correct.

Which of the following is (are) correct?   (d)
a) In Hebbian plasticity, the strength of a synaptic connection is increased whenever the presynaptic and postsynaptic cells are active together;
b) If a neuron receives Hebbian synaptic connections from other neurons, then it will strengthen connections with a limited subset of the presynaptic cells that all have correlated behaviors and will reduce the
weights of all the other connections to 0;
c) An excitatory lateral connection from cell A to cell B will move the afferent group of cell B towards the afferent group of cell A, until the two afferent groups become identical;
d) All of the above are correct;
e) None of the above is correct.



Lecture 16, October 15, 2002:   HEBBIAN SYNAPTIC PLASTICITY

It has been demonstrated in experimental studies in the cerebral cortex that when a presynaptic cell and the postsynaptic cell both are very active during some short period of time, then their
synapse increases its efficacy (or weight).  This increase is very long-lasting (maybe even permanent) and it is called LONG-TERM POTENTIATION (LTP).

If the presynaptic cell is very active during some short period of time, but the postsynaptic cell is only weakly active, then their synapse decreases its efficacy.  This decrease is also very
long-lasting (maybe even permanent) and it is called LONG-TERM DEPRESSION (LTD).

In general, there is an expression: “cells that fire together, wire together.”  Or, more correlated the behaviors of two cells, stronger their connection.

This idea was originally proposed by Donald Hebb in 1949 and, consequently, this type of synaptic plasticity (where correlated cells get stronger connections) is called HEBBIAN SYNAPTIC
PLASTICITY.  The rule that governs the relationship between synaptic weight and activities of the pre- and postsynaptic cells is called HEBBIAN RULE.

Currently, there are a number of mathematical formulations of the Hebbian rule.  None of them can fully account for all the different experimental results, so they all should be viewed as
approximations of the real rule that operates in the cortex.

One of the basic versions of Hebbian rule is COVARIANCE RULE:
         At time t, the change in the strength of connection between source cell s and target cell t is computed as
         Dw = (As – As) * (At – At) * RL,
                where As is the activity of the source cell (e.g., instantaneous firing rate),
           As is its average activity,
           At is the output activity of the target cell and
           At is its average activity.
                LR is “rate of learning” scaling constant.
This is “covariance” rule, because it computes across many stimuli the covariance in coincident activities of the pre- and postsynaptic cells.  Stronger the covariance (correlation in behaviors of
the two cells), stronger the connection.

A neuron has a limited capacity how many synapses it can maintain on its dendrites.  In other words, there is a top limit on the sum of all the connection weights a cell can receive from other
cells.  We will call this wmax.

To accommodate this limitation, we can extend the covariance rule this way:
        At time t, compute Dw of all the connections on the target cell -
Dwi = (AsiAsi) * (At – At) * RL
        Next compute tentative new weights -  w’i = wiDwi
        If the sum of all w’i (Sw’i) is less or equal wmax, then new weights are wi  = w’i
        If Sw’i  > wmax, then new weights are wi  = w’i * wmaxSw’i

This way, sum of weights will never exceed wmax.

Text:  pp. 680-681.



Lecture 17, October 17, 2002:   AFFERENT GROUP SELECTION

Hebbian rule makes presynaptic cells compete with each other for connection to the target cell:

(1)  Suppose a target cell had connections from only 2 source cells.  Then if these two presynaptic cells had similar behaviors (i.e., highly overlapping RFs), they will share the target cell equally and will give it
the behavior that is average of theirs.

(2)  If the two presynaptic cells had different behaviors (i.e., nonoverlapping or just minimally overlapping RFs), then they will fight for connections until one leaves and the other takes all.

(3)  If a neuron receives Hebbian synaptic connections from many other neurons, then the target cell will strengthen connections only with a limited subset of the source cells that all have correlated behaviors
(prominently overlapping RF profiles) and will reduce the weights of all the other connections to 0.  This subset – defined by having greater than 0 connection weights on the target cell – is the “AFFERENT
GROUP” of that target cell.
 

MODELING PROJECT #3:  AFFERENT GROUP FORMATION

HOMEWORK ASSIGNMENT #5

Write a computer program to implement Hebbian learning in the network set up in the previous modeling project, i.e., the network of one target cell that receives excitatory inputs from 40
source cells.  The source cells have somatotopically arranged RFs.
Task:  apply point stimuli to the receptor surface (skin) and adjust all the connections according to the Hebbian rule.
Goal:  develop a stable afferent group for the target cell.

Model parameters:    N = 40  - number of source cells.
                                RFR = 3  - receptive field radius.
D = 1  - spacing between receptive fields of neighboring neurons.
                                Cex = 5 - excitatory scaling constant
                                RL = 0.00001 - rate of learning by the connections

Initial weights of connections of source cells to the target cell, w(i), should be chosen randomly and then normalized so that Sw(i) = 1

Activity of source cell i is:  As(i) = 1 - (|S - RFC(i)| / RFR)
              where S - stimulus location on the receptor surface,
                        RFC(i) - receptive field of cell i.  RFC(i) is computed as RFC(i) = RFR + (i – 1) * D
            As is “instantaneous firing rate (or frequency)”; it is calculated as an inverse of time interval between successive action potentials.
                                If  As(i) < 0,  set  As(i) = 0.

The target cell is modeled as a point electrical circuit, made up of passive membrane conductance gm = 1 (E = -70) and net excitatory conductance Gex (E = 0).
Activity of the target cell is: DV = 70 * Gex / (Gex + gm)

In the steady state, Gex would be Gex = Cex * S w(i) As(i),  where w(i) is the connection weight of source cell i on the target cell.
To reflect temporal behaviors of ion channels,
                             Gex = (1 – 1/t) * Gex + (1/t) * Cex * S w(i) *As(i)
                                        Use t = 4.

The new feature of the model is that after each stimulus all the connection weights wi should be adjusted according to the Hebbian rule spelled out below.

Program flow:
Phase 1:    Set all the parameters (RFR, RFC(i), w(i), D, N, t, Cex).  New parameter is RL.
                Set average activity parameters of the source and target cells to 0: As(i) = 0 DV = 0
Phase 2:   Deliver 100000 point stimuli, picked in a random sequence anywhere on the entire receptor surface.
                For each stimulus location:
                              Compute activities of all the source cells, As(i)
                              Set Gex = 0
                              Do 20 time steps, updating Gex and DV.
                  Adjust connection weights:
                        - compute for all cells w’(i) = w(i) + RL * (As(i) – As(i)) * (DV – DV)
                        - if w’(i) < 0, set w’(i) = 0
                        - compute SUM = S w’(i)
                        - if SUM < 1, set SUM =1
                        - compute new values of connection weights:  w(i) = w’(i) / SUM
                        - update DV and As(i)DV = 0.99 * DV + 0.01 * DV
                                           As(i) = 0.99 * As(i) + 0.01 * As(i)

Phase 3:   After 100000 stimuli a local afferent group should form.
Show it by (1) plotting w(i) as a function of source cell number, i, and (2) plotting the RF profile of the target cell (i.e., plot DV as a function of stimulus location on the receptor surface).

Submit for grading:
        Brief description of the model
        The two plots
        Text of the program



Lectures 18-20, October 24-31, 2002:
 

LATERAL VS. AFFERENT CONNECTIONS

As we discussed in the previous lecture, if two cells, A and B, receive afferent connections from a set of source cells and these connections are Hebbian, then each target cell will strengthen
connections with some subset of the source cells that have similar behaviors, thus choosing these cells as its afferent group.  The two cells, A and B, will likely choose different subsets of the
source cells for their afferent groups.

Next, if the target cells are also interconnected by LATERAL connections, then these connections will act on the cells’ afferent groups, and the effect will depend on whether the lateral
connection is excitatory or inhibitory.

An excitatory lateral connection from cell A to cell B will move the afferent group of cell B towards the afferent group of cell A, until the two afferent groups become identical.

An inhibitory lateral connection from cell A to cell B will move the afferent group of cell B away from the afferent group of cell A, until the RFs of the two afferent groups do not overlap
anymore.

Thus, LATERAL EXCITATION = ATTRACTION of afferent groups.
 LATERAL INHIBITION = REPULSION of the afferent groups.
 

CORTICAL FUNCTIONAL ARCHITECTURE

Cerebral cortex is approximately 1.5 – 2 mm thick and is subdivided into 6 cortical layers.
Layer 1 is the topmost layer, it has almost no cell bodies, but dendrites and axons.
Layers 2-3 are called UPPER LAYERS
Layer 4 and bottom part of Layer 3 are called MIDDLE LAYERS (thus upper and middle layers overlap according to this terminology)
Layers 5 and 6 are called DEEP LAYERS

Layer 4 is the INPUT LAYER
Layers 2-3 and 5-6 are OUTPUT LAYERS

External (afferent) input comes to a cortical area to layer 4 (and that is why it is called “input” layer).
This afferent input comes from a thalamic nucleus associated with the particular cortical area and from layers 2-3 of the preceding cortical areas (unless this is a primary sensory cortical area).
Layer 4 in turn distributes input it receives to all other layers.
Layers 2-3 send their output to deep layers and to other cortical areas.
Layer 5 sends its output outside the cortex, to the brainstem and spinal cord.
Layer 6 sends its output back to the thalamic nucleus where this cortical locus got its afferent input, thus forming a feedback loop.

Excitatory neurons in layer 4 are called SPINY STELLATE CELLS.  They are the cells that receive afferent input and then distribute it radially among all other layers.
Axons of individual spiny stellate cells form a narrow bundle (they do not spread much sideway) and consequently they activate only a narrow column of cells in the upper and deep layers.  Thus
the afferent input is distributed preferentially vertically and much less horizontally.
As a result, cortical cells have more similar functional properties (e.g. RF location) in the vertical dimension than in the horizontal dimensions.  Going from one cell to the next in a cortical area,
RFs change much faster when going across the cortex horizontally than vertically.
That is, CORTEX IS ANISOTROPIC; it has COLUMNAR ORGANIZATION (i.e., more functional similarity vertically than horizontally).
In other words, a cortical area is organized in a form of CORTICAL COLUMNS.

Because cells in the radial (vertical) dimension have similar RFs, the topographic maps at all cortical depths are in register – there is a single topographic map for all cortical layers.
This topographic map is established in layer 4, because it is the input layer.

FUNCTION OF LAYER 4 IS TO ORGANIZE AFFERENT INPUTS TO A CORTICAL AREA.

Reading:  pp. 328-332, 335-337
 

 DEVELOPMENT OF CORTICAL TOPOGRAPHIC MAPS

General layout is genetically determined (see lecture3 #13-15), preserving topological relations of the source area in its projection onto the target area.
In addition there is fine-tuning of topographic maps in layer 4.
The mechanism of this fine-tuning is based on Hebbian synapses of afferent connections and lateral connections in layer 4, adjusting afferent connections to produce an optimal map.

MEXICAN HAT refers to patterns of lateral connections in a neural network in which each node of the network has excitatory lateral connections with its closest neighbors, but inhibitory lateral
connections with more distant neighbors.

Role of the Mexican Hat pattern of lateral connections in layer 4:
lateral interconnections move afferent groups of spiny stellate cells in layer 4, fine-tuning topographic map there.
Mexican Hat spreads RFs evenly throughout the input space, making sure that no region of the input space (e.g., a skin region or a region of retina) is not processed by the cortical area.
If some regions of the input space are used more than others and receive a rich variety of stimulus patterns (e.g., hands of a pianist or a surgeon), Mexican hat will act to devote more cortical territory to
processing inputs from those regions.  That is, in the cortical topographic map these regions will be increased in area (while underused regions of the input space will correspondingly shrink in size). Also, sizes
of RFs in more used regions will become smaller, while in the underused regions RFs will become larger.

An extreme case of underuse of  an input space region  involves somatosensory system – when a part of a limb is amputated. In this case, the cortical regions that originally received input from
this part of the body will gradually establish new afferent connections with neighboring surviving parts of the limb.

To summarize, the task of Mexican hat pattern of lateral connections in layer 4 is to provide an efficient allocation of cortical information-processing resources according to the individual’s
specific, idiosyncratic behavioral needs.

Reading:  pp. 328-332, 335-337

INVERTED MEXICAN HAT pattern of lateral connections - in this type of pattern of lateral connections, each node (locus) of the network has inhibitory lateral connections with its closest neighbors, but excitatory lateral connections with more distant neighbors.

Inverted Mexican hat will drive immediate neighbors to move their afferent groups away from each other (due to their inhibitory interconnections), but will make them stay close to afferent
groups of the farther neighbors (due to their excitatory interconnections).  As a result, RFs of cells in a local cortical region will become SHUFFLED – they will stay in the same local region of
the input space, but will overlap only minimally among closest cells.

The place for Inverted Mexican hat in layer 4:
cells in the cerebral cortex are organized into MINICOLUMNS.  Each minicolumn is a radially oriented cord of neurons extending from the bottom (white matter/layer 6 border) to the top (layer1/layer 2
border).  A minicolumn is approximately 0.05 mm in diameter and is essentially one-cell wide cortical column.

Neurons within a minicolumn have very similar functional properties (very similar RFs), but adjacent minicolumns have dissimilar functional properties (their RFs overlap only minimally).  As a
result, neurons in adjacent minicolumns are only very weakly correlated in their behaviors. In local cortical regions RFs of minicolumns appear shuffled – across such local regions RFs shift back
and forth in seemingly random directions.  It is only on a larger spatial scale – looking across larger cortical distances – that the orderly topographic map becomes apparent.
Thus, a cortical area has a topographic map of the input space, but this map looks noisy on a fine spatial scale.

The likely mechanism responsible for shuffling RFs among local groups of minicolumns is an Inverted Mexican hat pattern of lateral connections:  adjacent minicolumns inhibit each other, but
excite more distant minicolumns.

The purpose of Inverted Mexican hat – to provide local cortical regions (groups of minicolumns) with diverse information about a local region of the input space.

Overall, the pattern of lateral connections in the cortical input layer (layer 4) apparently is a combination of a small-scale Inverted Mexican hat and larger-scale Mexican hat.
That is, each minicolumn inhibits its immediate neighbors, excites 1-2 next neighbors, and again inhibits more distant minicolumns.



Lecture 21, November 5, 2002:   ORGANIZATION OF CORTICAL OUTPUT LAYERS

Approximately 80% of cortical cells are PYRAMIDAL cells.  These are excitatory cells.
The other 20% belong to several different cell classes:
- SPINY STELLATE cells; excitatory cells located in layer 4
- CHANDELIER cells; inhibitory cells in upper and deep layers, with synaptic connections on the initial segments of axons of pyramidal cells, and therefore in position to exert inhibition most
effectively
- BASKET cells; inhibitory cells in all layers, with synapses on somata and dendrites of pyramidal and other basket cells
- DOUBLE BOUQUET cells; inhibitory cells located in layer 2 and top layer 3, with synapses on dendrites of pyramidal and spiny stellate cells
- BIPOLAR cells; inhibitory cells
- SPIDER WEB cells; inhibitory cells in layers 2 and 4

(Note: the above list of targets of connections of different types of cells is not complete; it only mentions the most notable ones)

Cells in the output layers have dense and extensive connections with each other.  Inhibitory cells send their axons for only short distances horizontally, typically less than 0.3-0.5 mm.
Basket cells are an exception: they can spread their axons for up to 1 mm away from the soma.
Pyramidal cells have a much more extensive horizontal spread of axons.  Each pyramidal cell makes a lot of synapses locally (within approx. 0.3-0.5 mm cortical column), but it also sends axon
branches horizontally for several mm (2-8) away from the soma and forms sparse connections over such wide cortical territories.  These connections are called LONG-RANGE
HORIZONTAL CONNECTIONS.
Pyramidal cells, of course, also send some axon branches outside their cortical area (to other cortical areas, or brainstem, or thalamus; see lecture 15).

Thus, cortical columns separated by 1 mm or more in a cortical area are linked by excitatory connections (of pyramidal cells) exclusively.  However, because these connections are made on
both excitatory cells and inhibitory cells, their net effect on the target cortical column can be either excitatory or inhibitory (depending on the relative strengths of these connections).  In fact,
inhibitory cells are more easily excitable than pyramidal cells and as a result long-range horizontal connections evoke initial excitation in the target column, quickly followed by longer period of
inhibition. This sequence of excitation-inhibition is produced by strong lateral input.  When the lateral input is weak, it does not activate inhibitory cells sufficiently and as a result it evokes only
excitatory response in the target column.



 Lecture 22, November 7, 2002:   CORTICAL NETWORK AS A NONLINEAR DYNAMICAL SYSTEM

Before we turn to information processing carried out in the output cortical layers, we should consider them as a NONLINEAR DYNAMICAL SYSTEM.  The reason is that dynamical systems (i.e., sets of
interacting elements of whatever nature) that are described by nonlinear equations have a very strong tendency for generating complex dynamical behaviors, loosely referred to as CHAOTIC DYNAMICS.
Henri Poincare was the first to recognize this feature of nonlinear dynamical systems 100 years ago, but a more systematic study of such dynamics started in 1960s with work by Edward Lorenz.

From these studies we know that even very simple dynamical systems can be unstable and exhibit complex dynamical behaviors; and more structurally complex systems are even more so.
Cortical network structurally is a very complex dynamical system and can be expected to generate complex dynamical behaviors whether we like them or not.  Such “unplanned” behaviors are
called EMERGENT BEHAVIORS or EMERGENT PHENOMENA.

Dynamical behaviors can be displayed as TIME SERIES or PHASE-SPACE PLOTS

Phase-space plots can be constructed in a number of ways:
1)   Use one variable (e.g., activity of one cell at time t) and plot it against itself some fixed time later, Xt vs. Xt+Dt.
More generally, you can make an N-dimensional phase-space plot by plotting Xt vs. Xt+Dt1 vs Xt+Dt2 vs. …. Xt+Dt(N-1).

2)   Plot one variable vs. its first derivative (for a 2-D plot) or vs. first and second etc. derivatives for higher-dimensional plots.
For example, you can plot position of a pendulum against its velocity to produce 2-D phase-space plot.

3)   Use N different variables (e.g. simultaneous activities of 2 or more different cells) and plot them against each other to produce N-dimensional plot.

Regardless of a particular approach to producing a phase-space plot, the resulting graphs will look qualitatively the same (but not quantitatively): they will all show a single point, or a closed
loop, or a quasi-periodic figure, or chaotic plot.

STEADY-STATE DYNAMICS
In the steady state the dynamical system has reached its DYNAMICAL ATTRACTOR.
The attractor might be:
FIXED POINT (a stable single state, showing up as a point in phase-space plots)
LIMIT CYCLE, or PERIODIC (periodic oscillations of the system’s state, showing up as a closed loop in phase-space plots)
QUASI-PERIODIC (oscillations that look almost periodic, but not quite; this is due to oscillations having two or more frequencies the ratio of which is an IRRATIONAL number)
CHAOTIC (non-periodic fluctuations)

CHAOS is DETERMINISTIC DISORDER. Chaotic attractor is a nonperiodic attractor.
Chaotic dynamics is distinguished by its SENSITIVITY TO INITIAL CONDITIONS.  That is, chaotic attractor can be thought of as a ball of string of an infinite length (because it never
repeats its path) in a high-dimensional state space.  Two distant points on this string can be very close to each other in the space (because the string is folded), but if we travel along the string
starting from these two points, then the paths will gradually diverge and can bring us to very distant locations in the state space.  This is what is meant by “sensitivity to initial conditions” – even
very similar initial conditions lead to very different outcomes.  This is also why it is impossible to make predictions about future developments of chaotic dynamical systems.

For very nice source on chaos (with great graphics) see:   hypertextbook.com/chaos
Also, a very popular book on chaos written for non-scientists is:  Glieck (1988) Chaos: Making a New Science.  ISBN 0140092501

Modeling studies of cortical networks demonstrate that they can easily produce complex dynamical behaviors, including chaotic.  Depending on network parameters, they can have fixed-point,
periodic, quasi-periodic, or chaotic attractors.  Each and every parameter of the network (e.g., densities of excitatory and inhibitory connections, relative strengths of excitation and inhibition,
different balances of fast and slow transmitter-gated ion channels, afterhyperpolarization, stimulus strength, etc.) has an effect on the complexity of dynamics.  Some increase the complexity,
others decrease, yet others have a non-monotonic effect.

To show how the complexity of dynamics varies with a particular network parameter, use BIFURCATION PLOTS.
A bifurcation plot is a plot with the horizontal axis representing the values of the studied parameter, while the vertical axis represents a section through the phase space, e.g., the values of some
variable (such as activity of a particular cell) at which the first derivative of that variable is equal 0.

FRACTAL NATURE OF DYNAMICS:
the progression of complexity of dynamics from fixed-point to chaotic is not monotonic.
With even a tiny change in the controlling parameter, dynamics can change from chaotic to quasi-periodic or periodic, and back, and forth.
Thus, for example, two even very similar stimuli can have attractors of very different types.
What changes monotonically is the probability of having dynamics of certain type.

Based on modeling studies, cortical networks can readily generate chaotic dynamics.  But there are also opposite factors:
- complexity of dynamics is bounded, it does not grow with an increase in the structural complexity of the network beyond the initial.
- Stimulus strength reduces complexity of dynamics, so stronger the stimulus less chaotic the dynamics.
- Random noise (which is an unavoidable feature of biological systems) reduces the complexity of dynamics, converting chaotic dynamics to quasi-periodic-like ones.

Overall, it appears likely that cortical networks operate at the EDGE OF CHAOS.

TRANSIENT DYNAMICS
Cortical networks never reach their dynamical attractors, because they deal only with TRANSIENTS.  It is impossible to have steady-state dynamics in cortical networks, because of (1)
constant variability of sensory inputs, (2) adaptation in sensory receptors and in neurons in the CNS, and (3) presence of long-term processes in neurons.

Transients are much more complex than steady-state dynamics. A dynamical system might have a fixed-point attractor, but to get to it the system might have to go through a very complex,
chaotic-looking temporal process.

Although transients in cortical networks look very chaotic, they are quite ORDERLY in that there is an underlying WAVEFORM.  This underlying waveform is very stimulus-specific – even
small change in the stimulus can greatly change the shape of this waveform.

Conclusions
   We draw a number of lessons from our studies of model cortical networks.  First, one major source of dynamics in cortical networks is likely to be the sheer structural complexity of these
networks, regardless of specific details.  This should be sufficient for emergence of quasiperiodic or even chaotic dynamics, although it appears from our studies that such spurious dynamical
behaviors will be greatly constrained in their complexity.  This constraint is fortuitous, considering that a crucial requirement for perception, and thus for the cortex, is an ability to attend to some
details of the perceived sensory patterns and at the same time to ignore other, irrelevant details.  A high-dimensional dynamics with its great sensitivity to conditions – which would include
perceptually irrelevant ones – would present a major obstacle for such detail-invariant information processing.  In contrast, low-dimensional dynamics might offer a degree of sensitivity to
sensory input details that is optimal for our ability to discriminate among similar stimuli without being captives of irrelevant details.
   Second, spurious dynamics is likely to contribute many intriguing features to cortical stimulus-evoked behaviors.  We tend to expect specific mechanisms for specific dynamical behaviors, but
the presence of spurious dynamics should warn us that a clearly identifiable dynamical feature does not necessarily imply a clearly identifiable cause: the cause might be distributed – everywhere
and nowhere.
   Finally, spurious dynamics has great potential to contribute to cortical cells’ functional properties, constraining (and maybe in some cases expanding) information-representational capabilities of
cortical networks.  Some of these contributions might be functionally insignificant, others might be useful, and yet others might be detrimental and thus require cortical networks to develop special
mechanisms to counteract them.



Lecture 23, November 12, 2002:   ERROR-BACKPROPAGATION LEARNING

Neural networks can be set up to learn INPUT-TO-OUTPUT TRANFER FUNCTIONS.
That is, the network is given a set of input channels IN1 IN2 ….INn.  A training set is specified, in which for each particular input pattern (IN vector) there is a “desired” output, OUT (output
might be a single variable or a vector, but here we will focus on a single output).  Formally, OUT = f (IN1 IN2 ….INn) and f is called a TRANSFER FUNCTION.   The network’s task is to
learn to produce correct output for each input vector in the training set.  This is accomplished by presenting the network with a randomly chosen sequence of training input-output pairs,
computing the network’s responses and adjusting weights of connections in the network after each such presentation.

ERROR-CORRECTION LEARNING is a type of learning in neural networks where connection weights are adjusted as a function of error between the network’s desired and actual
outputs.  This is SUPERVISED LEARNING.

ERROR-BACKPROPAGATION LEARNING is a version of error-correction learning used in nonlinear multi-layered networks.

BACKPROP NETS can vary in number of input channels, number of layers of hidden units, and number of units in each hidden layer.

The power of backprop algorithm is that it can in principle learn any transfer function, given enough hidden layers and enough units in those layers.  However, in practice it takes more time to
learn more complex transfer functions, and this learning time grows very quickly for even moderately complex nonlinear functions to unacceptably long periods.  Also, the network might settle on a less than
optimal solution.

Homework Assignment #6

Modeling Project #4:  ERROR-BACKPROPAGATION (“BACKPROP”) NEURAL NETWORK

Write a computer program to implement Error Backpropagation Learning Algorithm in a backprop network made up of 2 input channels IN1 and IN2, one layer of 10 hidden units, and 1 output unit.

The network's task is to learn the relationship between activities of input channels IN1 and IN2 and the desired output OUTdesired.  An input channel's activity can be either 0 or 1.
This relationship is :  OUTdesired = IN1 exclusive-or IN2.
Thus there are only 4 possible input patterns:

IN1       IN2    OUTdesired
0           0             0
0           1             1
1           0             1
1           1             0

Activity of hidden unit i is:     Hi = tanh(Win1i * IN1 + Win2i * IN2)
tanh() is the hyperbolic tangent function   tanh(x) = (ex - e-x) / (ex + e-x)

Activity of the output unit is:  OUT = tanh( S Whi * Hi)

Assign initial weights to all connections RANDOMLY.
     -3 < Win <+3
     -0.4 < Wh < +0.4

Present 1000 input patterns chosen randomly among 4 possible ones.

For each input pattern compute all Hs, OUT, and then adjust connection weights according to these steps:
      (1)  Compute error:  ERROR = OUTdesired - OUT
      (2)  Compute error signal: d = ERROR * (1 – OUT2)
      (3)   Adjust hidden unit connections:   Whi = Whid * Hi * RLh
                      where RLh = 0.003 is rate of learning
      (4)  Backpropagate error signal to each hidden unit: did * Whi * (1- Hi2)
       (5)  Adjust input unit connections:   Winij = Winijdi * INi *RLin
                      where RLin = 6 is rate of learning
 
 

Submit for grading:   brief description of the project
                               plot of   | ERROR |  as a function of training trial #.
                               text of the program



 Lectures 24-26:  HOW CEREBRAL CORTEX MAKES SENSE OF THE OUTSIDE WORLD

1. Introduction

We live in a world that is to a large degree orderly. All aspects of our lives – the way our hands interact with objects or our balance and mobility are affected by the surface we stand on, what events of behavioral significance happen around us, what impacts we can make on things and events either directly or indirectly, our overall living conditions, etc. – all are to a large degree predictable by us and this enables us to have purposeful and successful behaviors.Mammals, in general, are very adept at discovering this order through personal learning. They are born with only rudimentary innate skills and understanding of the world, but in the course of interacting with their surroundings they gradually discover regularities that are hidden in spatiotemporal patterns of activities of their sensory receptors and acquire great expertise in dealing with the challenges and opportunities posed by their environments. This ability to discover hidden order is the brain’s most distinctive feature and the primary source of its intelligence.
Discovering hidden order is probably the most challenging task performed by the brain. Its demands are certain to be among the most fundamental determinants that shaped the brain’s functional design. In this paper we consider obstacles to learning environmental regularities, propose how these obstacles can be overcome and formulate a general computational approach to discovering deeply hidden regularities. We next suggest how this approach might be implemented in the neocortex. Finally, we conclude with how the neocortex might represent internally the discovered environmental order and how such an internal model of the outside world might guide interactions of an animal with its environment. 

2. The core idea: learning the identities of environmental factors 

2.1. What are the obstacles to learning regularities? 

Interdependence among environmental variables – arising from interactions among the constituents of the environment – makes it possible to infer (i.e., compute with some degree of accuracy) the state of one variable from the known states of some other variables.In other words, the variables have inferential relations with other variables. The question that we will consider first is how can a hypothetical observer discover as many such inferential relations in the observed environment as possible.
We will illustrate our considerations with an example of kitchen sinks and a prototypical, “toy” brain whose task is to learn the orderly nature of sinks through observation of different sinks in different states. To simplify matters, we will suppose that all sinks in our example receive water from two pipes and are controlled by the following five variables:HC indicates which of the two pipes carry hot and cold water (0 – the left pipe is hot and the right pipe is cold, 1 – vice versa), DIRL and DIRR are directions in which the left and the right knobs should be turned to open the pipes (0 – clockwise, 1 – counterclockwise), and KPL and KPR are positions of the two knobs.Sinks in different kitchens can have different DIRL, DIRR, and HC; KPL and KPR can vary across time in the same sink.
These five variables determine the flow of water through the two pipes:
and(1)

In turn, flow of water through the two pipes determines the total water outflow from the faucet and its temperature:

and(2)

The task of learning the orderly nature of sinks is defined explicitly as learning how to determine the state of each variable from the states of other variables.

Brains have only limited direct sensory access to things in the world.To reflect this situation, suppose our toy brain cannot see inside the pipes and therefore it does not know about the flow of water through them; i.e., variables FL and FR are not given to it by its sensors.As we can see from equations 1-2, having FL and FR makes relations among the variables quite simple.Not having FL and FR will make these relations more complex.Luckily, because FL and FR are implicit in the states of other variables, the relations among the remaining variables can still be defined without FL and FR.But these relations will be more complex.For example,

(3)

or

.(4)

And the more complex the relations, the more difficult (and, at some point, impossible) to learn them.

To draw a lesson from this example, a major source of difficulties in learning regularities (inferential relations) is not knowing about the existence of some of the factors involved in a regularity.This is a fundamental problem faced by real brains, considering that their sensory receptors reflect only very indirectly on the environmental variables that are directly relevant to behavior.If an environmental variable that plays an important role in some regularity is not among the variables known to an observer, but is reflected implicitly in the behaviors of some of the known variables, then the observer can still, in principle, learn the regularity.However, the regularity will now become more complex, involving all these extra variables with their implicit information about the missed key variable.More variables and, likely, more nonlinearities contributed by them will make the regularity more difficult to learn.Given that learning capabilities are not unlimited (no matter how intelligent is the observer), a regularity can thus easily be placed beyond the observer’s grasp.

This lesson illustrates an argument of Clark and Thornton (1997) about the necessity of representing inferentially important environmental variables explicitly: to learn regularities, it is crucial first to learn separately the identities of as many environmental factors contributing to those regularities as possible.Clark and Thornton call this “trading computation for representation.”In the case of kitchen sinks, our toy brain should learn about the flow of water through each pipe; i.e., it should learn the existence (and states) of variables FL and FR from observing behaviors of the known variables.Knowledge of FL and FR will simplify the toy brain’s task of learning regularities; i.e., how each variable relates to other variables.For example, instead of equations 3 and 4, it will need to learn simpler equations:

and (5)

But, how can our toy brain discover the existence of these inferentially important variables FL and FR?It will have to derive them from the known variables; i.e., it will have to learn to compute their states from the states of the known variables.

2.2.How to discover important environmental variables?

Hidden environmental variables that are involved in regularities, but are reflected only implicitly in the known variables, can be learned from the known variables by looking for what Barlow (1992) calls “suspicious coincidences” among sensory events.That is, we should look for “something” in one subset of the known variables that correlates with “something” in another such subset.These “somethings” are, mathematically, functions of some sorts over the two subsets of the known variables.Thus, to follow Barlow, we should look among various functions over various subsets of the known variables, searching for such pairs of functions over non-overlapping subsets of variables that will show correlated behaviors.Such different-but-correlated functions must have an external reason for their statistical interdependence, a causal source in the environment.This means that by identifying a pair of correlated functions we will, in effect, detect some causal factor operating in the environment that is responsible for this correlation (Craik, 1942; Barlow, 1992; Becker and Hinton, 1992; Phillips and Singer, 1997; Ryder and Favorov, 2001).

Such causal factors are very likely to have other effects in the observed environment, besides the ones that led to their recognition.And once we have learned to recognize a causal factor by one of its effects, it will now become easier to notice its other effects.First, it will become easier to learn those inferential relations among the variables that involve this factor. And second, it will also become easier to learn to recognize this factor by its other – subtler – manifestations.Finally, once we have learned to recognize a number of causal factors, we can now look for suspicious coincidences among them (i.e., among functions over these factors), thus discovering higher-order causal factors, etc.

Thus, if we apply this method of discovering inferentially useful variables to kitchen sinks, we will observe that the following two functions, and , correlate perfectly; i.e., The fact that these two functions correlate perfectly indicates that they compute a new variable that plays an important role in kitchen sinks.Of course, these two functions do not reveal the physical identity of the discovered environmental factor (it is, by the way, the flow of water in the left pipe, FL), but nevertheless variable X expresses the state of this factor and will simplify inferential relations among sink variables.

Thus, to summarize, we can find important, but hidden environmental variables from suspicious coincidences.The basic idea (Becker and Hinton, 1992; Phillips and Singer, 1997; Ryder and Favorov, 2001) is to search for suspicious coincidences among various functions over various subsets of the known variables, trying to find functions over non-overlapping subsets that will behave identically, or at least very similarly.Once we find such correlated functions, then we can use them as alternative estimates of the same, likely to be inferentially useful, environmental variable; e.g., 

 
2.3.How does neocortex find correlated functions?
We (Ryder and Favorov, 2001) have proposed that the search for correlated functions is performed in the neocortex by dendritic trees of individual pyramidal cells.Here we will summarize the main points of that proposal.The computational approach to finding correlated functions over different inputs has been developed by Becker and colleagues (refs.).The basic idea is to set up two or more computational modules that look at separate but related parts of the sensory input, and to make these modules teach each other to produce outputs that match as closely as possible.In this way the modules will learn to perform different operations on their different inputs so as to yield correlated outputs.
Becker’s basic concept forms one of the cornerstones of Phillips and Singer’s (1997) view of the cerebral cortex.Phillips and Singer propose that contextual (i.e., lateral, as oppose to afferent) inputs to cortical cells guide them to tune to those stimulus features in their receptive fields that are predictably related to the context in which they occur.In this way, by discovering predictable relations between different inputs (within vs. outside their receptive fields), cortical cells discover objectively important variables in the external world.Phillips and Singer (1997) offered a possible mechanism for how such tuning might be accomplished.Unfortunately, that mechanism is limited in its practical utility due to its inability to search for correlations among nonlinear functions, although those are the functions that in the real world are most likely to express environmental factors in terms of other environmental factors.
Phillips and Singer’s focus on individual pyramidal cells as discoverers of correlated functions is very appealing for several reasons.First, placing the mechanism in the dendritic trees of individual cells (rather than requiring multi-cellular modules) expands by orders of magnitude the number of inferentially useful variables that the brain will be able to discover in the environment and makes this mechanism much more practical biologically.

Next, pyramidal cells have 5-8 principal dendrites growing from the soma, including 4-7 basal dendrites, and the apical dendrite with its side branches (Feldman, 1984). Each principal dendrite sprouts an elaborate, tree-like pattern of branches and is capable of complex forms of integration of synaptic inputs it receives from other neurons (Mel, 1994; Häusser et al., 2000; Segev and London, 2000).Synaptic learning in dendrites is controlled both by presynaptic activity and by the output activity of the postsynaptic cell; the latter is signaled to each synapse by spikes that are backpropagated from the soma up through each dendrite (Markram et al., 1997b; Stuart et al., 1997).Thus, through its contribution to the cell’s output, each principal dendrite can influence the strengths of synaptic connections on other of the cell’s dendrites.Due to the Hebbian nature of synaptic plasticity (e.g., Singer 1995; Paulsen and Sejnowski, 2000), the effect of this influence will be for each principal dendrite to drive the other dendrites of the same cell to learn to behave the way it does.In other words, each principal dendrite will teach the other dendrites of the same cell to behave the way it does, while also learning from them. Through this mutual teaching and learning, all the principal dendrites in a cell should learn to produce correlated outputs, or “speak with a single voice.”

Next, the different principal dendrites of the same pyramidal cell receive different connections and consequently are exposed to different sources of information about the environment.For example, neighboring pyramidal cells make synaptic connections preferentially on the basal dendrites (Markram et al., 1997a), whereas more distant cells, including ones several millimeters away in the same cortical area, terminate preferentially on the apical dendrites (McGuire et al., 1991).Another system of connections, the feedback connections from higher cortical areas, terminate preferentially on yet another part of the dendritic tree, i.e., on the terminal tuft of the apical dendrite, located in layer 1 (Cauller, 1995).Yet another source of differences in inputs to different principal dendrites of the same cell is cortical topographic organization.Across a cortical area, the functional properties of cells change very quickly: even adjacent neurons carry in common less than 20% of stimulus-related information (Gawne et al. 1996; Favorov and Kelly, 1996).Basal principal dendrites extend in all directions away from the soma and thus spread into functionally different cortical domains. As a result, these dendrites sample different loci of the cortical topographic map and are exposed to different aspects of the cortical representation of the environment (Malach, 1994).

Thus it seems that the principal dendrites of pyramidal cells are well set up for finding different-but-correlated functions: they receive different sensory inputs and they are made to find such functions over these inputs that would behave similarly, if not identically.And by discovering correlated functions over their different subsets of sensory variables, dendrites will tune the cell to the environmental factor responsible for this correlation.

The final consideration concerns the ability of principal dendrites to teach each other nonlinear functions.Cortical synaptic plasticity appears to be Hebbian (e.g., Singer, 1995; Paulsen and Sejnowski, 2000), but the current theoretical formulations of this rule (such as, for example, Grossberg’s, 1974, star rules; Sejnowski’s, 1977, covariance rule; the BCM rule of Bienenstock et al., 1982; etc.) all are basically incapable of learning the most useful, nonlinear functions (e.g., Willshaw and Dayan, 1990; Hancock et al., 1991; Phillips and Singer, 1997).This limitation would greatly reduce the ability of pyramidal cells to find functions that would correlate with each other.However, we believe the problem is not that the synaptic rule used in the neocortex is weak, but that the current theoretical formulations do not yet capture it fully.The most recent experimental findings indeed demonstrate that neocortical synaptic plasticity is a much more complex, multifactorial phenomenon than is reflected in current formalisms (e.g., Häusser et al., 2000; Segev and London, 2000), suggesting that the neocortical synaptic rule has yet to be recognized capabilities.Based on the functional considerations discussed here, we expect that the synaptic rule will turn out to endow principal dendrites with the capacity to teach one another simple or even moderately complex nonlinear input-to-output transfer functions. This is something that ought to be explored experimentally.A promising step in this direction is a recent demonstration that LTP and LTD protocols in neocortical slices lead not only to changes in synaptic efficacies, but also to changes in nonlinear integrating properties of the engaged dendrites.

Thus, to conclude, all the considerations reviewed above point to individual pyramidal cells – and more specifically, their principal dendrites – as the most likely candidates for performing the task of finding correlated functions over different sets of environmental variables.It remains to be seen how complex are the functions that the principal dendrites of pyramidal cells are capable of teaching each other. We predict that they should be capable of teaching each other to perform nonlinear functions on their synaptic inputs.In this regard we deviate from Phillips and Singer (1997).In another substantial deviation from their idea, we propose that the inputs to a cell are divided into functional sets (for purposes of mutual teaching) by virtue of being located on different principal dendrites of a cell, and not necessarily by whether they are contextual or come from the cell’s receptive field.We named this conceptual model of the pyramidal cell ‘SINBAD.’ This name is an acronym that stands for a Set of INteracting BAckpropagating Dendrites, where “backpropagating” refers to the signal that is sent back along the dendrites from the soma, affecting synaptic plasticity.

2.4.How to pick sets of variables capable of yielding correlated functions?

So far in our considerations we took for granted that different principal dendrites of the same SINBAD cell will happen to have access to related sets of environmental variables; i.e., sets without any variables in common, but with implicit information about the same hidden environmental variable and therefore capable of producing correlated functions.Ideally, these sets should be chosen by some procedure that will maximize the chances that the sets will be related, while minimizing the number of variables in each set.The most obvious approach is to give a cell local inputs, taking advantage of the fact that most of regularities in natural environments (or, at least, most of regularities that we are able to appreciate perceptually) are local in one way or another.For example, lower-order regularities in peripheral input patterns involve environmental conditions in close spatial proximity to each other; consequently, exposing different dendrites of a pyramidal cell in primary sensory cortex to raw information from distant spatial locations would be useless.In agreement with this observation, afferent connections to cortical areas do not all contact each and every cell but have clear topographic organization (e.g., body maps in somatosensory and motor cortices, retinotopic maps in visual cortex, etc.).These maps are created in middle cortical layers by a host of genetic and epigenetic mechanisms (e.g., von der Malsburg and Singer, 1988).From the viewpoint advanced here, we expect that the mechanisms that control perinatal development and adult maintenance of cortical topographic maps are designed to supply each cortical neighborhood with limited but functionally related information in order to improve cells’ chances of finding important environmental variables (Barlow, 1986). 

To suggest one idea, as we reviewed elsewhere (Favorov and Kelly, 1996), when receptive fields of cortical cells are considered in toto, in all their dimensions, neighboring neurons typically have little in common - a stimulus which is effective in driving one cell will frequently be much less effective in driving its neighbor.This prominent local receptive field diversity is constrained, however, in the radial cortical dimension. Cells that make up individual radially oriented minicolumns – 0.05mm diameter cords of cells extending across all cortical layers (Mountcastle, 1978) – have similar receptive field properties (Abeles and Goldstein, 1970; Hubel and Wiesel, 1974; Albus, 1975, Merzenich et al., 1981; Favorov and Whitsel, 1988; Favorov and Diamond, 1990; summarized in Favorov and Kelly, 1996).If neighboring neurons have contrasting functional properties, they are likely to belong to different minicolumns. 

Thus it appears that local groups of minicolumns bring together a variety of different, but related sensory information concerning a local region of the stimulus space, with adjacent minicolumns tuned to extract only minimally redundant information about what takes place in that stimulus space.Because this information comes from a local region of the stimulus space, it is likely to be rich in regularities reflecting the orderly features of the outside world.Thus, it appears that local cortical neighborhoods create local informational environments enriched in regularity, but low in redundancy (for a possible mechanism responsible for differences among neighboring minicolumns, see Favorov and Kelly, 1994a,b).Such local cortical environments are exactly the right informational environments for pyramidal cells, to be mined by them in their search for correlated functions.Each dendrite of a pyramidal cell will be extended through a functionally distinct group of minicolumns, exposing it to a unique combination of afferent information (Malach, 1994).Principal dendrites of the same cell will thus have an opportunity to find correlated functions over different environmental variables, tuning the cell to the underlying environmental factor. 

The distribution of input connections among principal dendrites of a cell can also be improved by trial-and-error rearrangement of connections on dendrites, if the dendrites fail to find any correlated functions.After a period of unsuccessful learning attempts, some randomly chosen connections might be dropped, while other new connections might be added.This would involve fine remodeling of dendrites and sprouting of axon collaterals, something that takes place even in the adult cortex (Quartz and Sejnowski, 1997).Dendrites can continue to “experiment” with their input connections until they find correlated functions over their inputs.This idea is closely related to “constructivist manifesto” of Quartz and Sejnowski (1997), whose main thesis is that cortical networks start small and then gradually add more structures (connections and integrative units in a form of dendritic branches) in response to specific computational demands posed by each network’s particular sensory environment.

3.An inferential model: the cortical representation of the world’s order

We now turn to the question of how a network of SINBAD neurons can learn the order inherent in the environment.The last section described how a SINBAD network can greatly expand the repertoire of variables with which it represents the environment, by deriving new variables from the original, sensory receptor set and from the already derived ones.The purpose of deriving new variables is to uncover the more prominent among the factors that operate in the observed environment.By identifying and representing such factors explicitly as environmental variables, the network will make it easier to learn environmental regularities involving those factors.

To describe how this works, suppose we start with a set of original variables that are only distantly related to each other.We then derive new variables from the original ones.As prominent environmental factors, these new variables have inferential significance to other variables and thus will provide inferential links between the variables used to determine their states, and the variables whose states can be inferred from them.Thus, by placing the newly derived variables in-between the original ones, we will break down the complex inferential relations among the original variables into simpler – and easier to learn – inferential relations of the original variables with the derived ones, and via those with each other.More inferentially significant variables we add to our repertoire (deriving them from the original and the already derived ones), more of the distant inferential relations will be broken down into simpler and easier to learn relations.

Thus, we should aim to derive as many variables as we can, and learn as many ways to infer each variable from the other ones as we can.By thus expressing each variable in many different ways in terms of other variables, which in turn are expressed in terms of yet other variables, etc., we will construct a rich web of inferential relations.In the ideal case, this web (which is grounded in the causal structure of the observed environment) will link each variable either directly or via intermediaries to every other variable.

Such learning of inferential relations can be accomplished in the cortex by making use of lateral connections among cells within each cortical area and feedback connections from higher-level cortical areas.Via these connections pyramidal cells will have access to the variables derived by other cells in the same and in the higher cortical areas.By linking pyramidal cells (directly and also via inhibitory cells) and by placing these connections on the cells’ dendrites, we will give principal dendrites additional information to use in their learning how to predict outputs of the other dendrites of the same cell.This way, each pyramidal cell will learn to determine the state of the environmental variable it represents from many alternative and complementary lines of evidence, some direct, others circumstantial.The cortex, as a whole, will capture the web of inferential relations among the original and derived environmental variables in its pattern of ascending, lateral and feedback connections.

In this web all the discovered inferential relations will be tied together into a single functional entity – an inferential model of the observed environment.The basic mode of operation of such an inferential model is what might be called “filling-in missing parts of a picture.”The picture is the model’s representation of the state and processes in the observed environment, cast in terms of the observer-chosen variables (original and derived).Faced with any particular situation, the observer will obtain information about some of the environmental variables from the sensors, thus establishing the picture’s factual foundation.These variables will “fill-in,” via their known inferential relations, the states of some other variables, adding new details to the emerging picture of the situation or clarifying some fuzzy ones.These new details might make it possible to infer the states of some other variables, which in turn might make it possible to infer yet more variables, and so on, thus gradually elaborating the picture of the situation with more and more details inferred flexibly and opportunistically from whatever circumstantial information is at hand.

If the observer itself is active in the observed environment, then its own needs, desires, intentions, and motor actions become important factors affecting the environment.Consequently, these factors will be identified as the environmental variables by the observer and will be incorporated into the web of inferential relations.In this case, the observer becomes a part of its picture of the observed environment, making its actions inferable from what is present and what is desired.Overall, the web of inferential relations can be used to infer what has happened, is happening, will happen, or what should be done, i.e., how to reach the desired goals.

Developing and using such an inferential model must be the brain’s central function.Indeed, at some fundamental level the work that the brain performs can be summarized as obtaining from its senses information about an environmental situation it faces and using its knowledge of inferential relations in the environment to expand the mental representation of this situation, adding more and more details about its past, its present, and its future, including its own actions, which are then translated into actual physical actions in the environment.
 

Cortico-thalamic feedback

Pyramidal cells in cortical layer 6 send connections back to the thalamus.  This feedback system of corticothalamic connections implements Mumford’s (1991) idea of the thalamus being used by the cortex as a “blackboard,” on which the cortex draws its interpretation of the attended subject.  In our use of this idea, the web of inferential relations learned by the cortical network acts as an inferential model of the outside world, and this internal model projects its picture of the outside world back on the thalamus, so that it can be returned again to the cortex for another pass of inferential adjustment and elaboration, and so on.  This will enable the cortical inferential model to fill-in holes, when they happen, in the raw picture of the world that the thalamus receives from its sensory channels.

Summary of functions of major components of thalamo-cortical network:

THALAMIC NUCLEUS - relays environmental variables from sensory receptors to cortical input layer (layer 4)

CORTICAL INPUT LAYER - organizes sensory inputs and delivers them to dendrites of pyramidal cells in the output layers

BASAL DENDRITES OF PYRAMIDAL CELLS - tune pyramidal cells to new orderly environmental features (i.e., inferentially useful environmental variables)

LATERAL CONNECTIONS ON APICAL DENDRITES - learn inferential relations among environmental variables

CORTICO-THALAMIC FEEDBACK - disambiguates and fills in missing sensory information in thalamus

HIGHER-ORDER CORTICAL AREAS - pyramidal cells there tune to more complex orderly features of the environment and learn more complex inferential relations

CORTICO-CORTICAL FEEDBACK - disambiguates and fills in missing information in lower-level cortical areas, making use of deeper understanding of the situation by the higher-level cortical areas


Lecture 27:  VISUAL SYSTEM
We will use visual system for an illustration of general principles. 

Cortex deals with INFORMATION.  Information comes, ultimately, from SENSORY RECEPTORS.  In visual system, receptors are called PHOTORECEPTORS.  There are 2 principal
classes of photoreceptors: CONES (for color vision) and RODS (for grayscale vision).  There are 3 types of cones, sensitive to different ranges of light wavelengths.  One is sensitive to red,
another to green, the third to blue colors.  The general lesson for all senses is that the nervous system employs MULTIPLE SUBMODALITIES of receptors within any given sense (visual,
auditory, olfactory, etc.). 

SENSORY PATHWAYS carry information from receptors to the cortex.  In visual system, the pathway consists of:  receptors -->  BIPOLAR CELLS  --> GANGLION CELLS  --> LGN
CELLS  --> CORTEX 

Photoreceptors, bipolar cells, and ganglion cells are all located in retina, while LGN is Lateral Geniculate Nucleus of Thalamus.  (In general, for other sensory systems, the cells in the pathway to
cortex are located in spinal cord or in brainstem.) 

The pathway cells perform SUBCORTICAL PREPROCESSING of the raw inputs.  In visual system, ganglion cells have circular RFs with center and opposite surround.  These RFs can be
either ON-CENTER or OFF-CENTER RFs.  The function of these RFs is CONTRAST ENHANCEMENT (sharpening lines, edges in visual images).
The functional meaning of subcortical cells – they represent original environmental  variables.  Thus, each thalamic cell represents one raw, or original, variable. 

INPUT LAYER 4 OF V1 (primary visual cortical area)
It takes connections from thalamic nucleus (LGN) and organizes inputs topographically. In V1, the TOPOGRAPHIC MAP is RETINOTOPIC WITH LOCAL RF SHUFFLING.
This map was developed originally due to Hebbian plasticity and Mexican/Inverted-Mexican Hat pattern of lateral connections in layer 4. 

The basic purpose of layer 4 topographic map is to create cortical neighborhoods enriched in different, but functionally related variables. 

Cells in layer 4 of V1 have “SIMPLE” RFs – small elongated RFs made up elongated central region that is flanked by opposite sign regions.  As a result, cells respond selectively to  lines or edges of particular orientations in particular locations in the visual field. 

Famous features of topographic organization of layer 4 of V1:  OCULAR DOMINANCE COLUMNS and ORIENTATION COLUMNS. 

V1 OUTPUT LAYERS
The main cell type in output layers is PYRAMIDAL CELL, whose function is to discover causal factors operating in the outside world and learn inferential relations among them. 

Knowledge of these inferential relations can be used, for example, to extract weak or noisy signals, to achieve translationally-, rotationally-, and size-invariant object recognition, to detect
configuration changes in the moving objects, to infer object features (e.g., contours) not visible directly. 

 In visual cortex, V1, cells in output layers develop “COMPLEX” RFs – a cell will respond selectively to lines or edges of a very particular orientation but in a range of locations
in the visual field.  Cells also develop preferences for stimuli  at particular distances from the eyes and also for particular directions of stimulus motion across the visual field. 

V2 AND HIGHER CORTICAL AREAS
Pyramidal cells in the upper layers of V1 project to the secondary visual cortical area, V2, and cells there project to yet higher areas, etc.  These projections terminate in the input layer, layer 4,
where these inputs get organized topographically (just as it was done in layer 4 of V1) for use by pyramidal cells in those areas.  These higher cortical areas will derive higher-order
variables, revealing deeper causal forces operating in the environment.  Knowledge of such high-order variables is crucial for our ability to generate successful behaviors, because
regularities among natural phenomena that are most useful behaviorally operate among features (i.e., variables)  of the environment that are reflected only very implicitly in the spatiotemporal
patterns of sensory receptor activities.
Two of the most important tasks facing the visual system, and all other sensory systems, are FIGURE-GROUND SEPARATION and INTERPRETATION of the seen images. 

Reading:  pp. 387-398 (perception), and optionally, 407-410, 416-419 (retina),  425-445 (organization of visual cortex). 



FINAL EXAM

December 3, TUESDAY, 1-4 pm in the regular classroom (portable). 

Subject:  lectures 16-27

Hebbian learning
Afferent groups
Lateral interactions among afferent groups
Cortical architecture
Topographic map development
Cortical dynamics
Error backpropagation learning
Learning inferential relations
Visual system

Exam will have 2 parts: multiple-choice questions (75% of the grade) and a connectional diagram drawing with an explanation of the parts and their functions (25% of the grade). 

Possible diagrams:
Pattern of lateral connections in the input cortical layer (Mexican Hat + Inverted Mexican Hat)
Cortical architecture (connections among layers, thalamus, and among cortical areas)
Error backpropagation network
Cortical functional organization (in terms of learning  environmental variables and inferential relations) 


 
                            FINAL EXAM (example)

PART 1  (75% of the total)

1.  Which one of the following statements is correct?
(a) A typical synapse in the cerebral cortex decreases its efficacy when the presynaptic cell and the postsynaptic cell are both very active during some short period of time.
(b) A typical synapse in the cerebral cortex does not change its efficacy when the presynaptic cell is very active during some short period of time, but the postsynaptic cell is only weakly active.
(c) A long-lasting increase in the efficacy of an excitatory synaptic connection is called LONG-TERM POTENTIATION (LTP).
(d) A short-lasting decrease in the efficacy of an excitatory synaptic connection is called LONG-TERM DEPRESSION (LTD).
(e) WOOLSEAN SYNAPTIC PLASTICITY can be summarized by the following expression:  “cells that fire together, wire together.”

2.  Which one of the following statements is correct?
(a) HEBBIAN SYNAPTIC PLASTICITY is the type of synaptic plasticity in which correlated cells weaken their connections.
(b) One of the basic mathematical formulations of the Hebbian rule is called CONTRAVARIANCE RULE.
(c) According to the Covariance Rule, the change in the strength of a connection between source cell s and target cell t is computed as  Dw = (As– As) * (At – At) * LR, where As is the
activity of the source cell, As is its average activity, At is the output activity of the target cell, At is its average activity, and LR is a scaling constant.
(d) A neuron has a limited capacity how many synapses it can maintain on its dendrites.
(e) The Covariance Rule will not raise the weight of a synaptic connection above 1.

3.  Which one of the following statements is correct?
(a) Hebbian rule makes presynaptic cells compete with each other for connection to the target cell.
(b) If a target cell had Hebbian connections from only 2 source cells and these two presynaptic cells had very similar behaviors (i.e., highly overlapping RFs), then both source cells will loose
their connections with that target cell.
(c) If a target cell had Hebbian connections from only 2 source cells and these two presynaptic cells had different behaviors (i.e., nonoverlapping or just minimally overlapping RFs), then both
source cells will develop equally strong connections with the target cell.
(d) If a neuron receives Hebbian synaptic connections from many source neurons, then it will strengthen connections only with one of these source cells and will reduce the weights of all the
other connections to 0.
(e) The group of cells that has connections with a given target cell is called the “INPUT GROUP” of that target cell.

4.  Which one of the following statements is correct?
(a) An excitatory LATERAL connection from cell A to cell B will move the afferent group of cell A towards the afferent group of cell B, until the two afferent groups become identical.
(b) An inhibitory LATERAL connection from cell A to cell B will move the afferent group of cell A away from the afferent group of cell B, until the RFs of the two afferent groups do not overlap
anymore.
(c) If two cells are interconnected by excitatory LATERAL connections, then these connections will act on the cells’ afferent groups, moving them towards each other.
(d) If two cells are interconnected by inhibitory LATERAL connections, then these connections will not have any effect on the cells’ afferent groups.
(e) A connection of a thalamic cell with a cortical cell is an example of a lateral connection.

 5.  Which one of the following statements is correct?
(a) Cerebral cortex contains 6 cortical layers.
(b) Layers 4-5 are called UPPER LAYERS.
(c) Layers 4-5 are called DEEP LAYERS.
(d) Layers 1-3 are the INPUT LAYERS.
(e) Layers 4-5 are the OUTPUT LAYERS

6.  Which one of the following statements is correct?
(a) External (afferent) input comes to a cortical area to layer 1 and that is why it is called the “input” layer.
(b) The afferent input to a cortical area comes from a thalamic nucleus associated with that cortical area and from layers 2-3 of the preceding cortical areas (unless this is a primary sensory
cortical area).
(c) Layers 5-6 send their output to deep layers and to other cortical areas.
(d) Layer 6 sends its output outside the cortex, to the brainstem and spinal cord.
(e) Layer 2 sends its output back to the thalamic nucleus where this cortical locus got its afferent input, thus forming a feedback loop.

7.  Which one of the following statements is correct?
(a) Excitatory neurons in layer 4 are called BASKET CELLS.
(b) Axons of individual spiny stellate cells in the input layer form a narrow radially-oriented bundle (they do not spread much sideway) and consequently they activate only a narrow radial
column of cells in the upper and deep layers.
(c) Afferent input to a cortical site is distributed uniformly in all directions.
(d) Cortical cells have more similar functional properties (e.g. RF location) in the horizontal cortical dimensions than in the vertical dimension.
(e) Cerebral cortex does not have a COLUMNAR ORGANIZATION.

8.  Which one of the following statements is correct?
(a) Inhibitory cells make up the majority of cells in the cerebral cortex.
(b) PYRAMIDAL cells in the cerebral cortex make inhibitory synaptic connections.
(c) Cells in the cortical OUTPUT LAYERS have very few connections with each other.
(d) Inhibitory cells in the cortical OUTPUT LAYERS send their axons for long (several millimeters) distances horizontally.
(e) Pyramidal cells send axon branches horizontally for several mm (2-8) away from the soma and form connections over such wide cortical territories.

9.  Which one of the following statements is correct?
(a) Pyramidal cells do not send axons outside their cortical area.
(b) Cortical columns separated by 1 mm or more in a cortical area are linked exclusively by inhibitory synaptic connections of pyramidal cells.
(c) Lateral connections in the output layers are made only on the excitatory cells and not on the inhibitory cells.
(d) Inhibitory cells are more easily excitable than pyramidal cells and as a result long-range horizontal connections evoke initial excitation in the target column, quickly followed by longer period
of inhibition.
(e) There is no such thing as a “cortical column.”

10.  Which one of the following statements is WRONG?
(a) General layout of cortical topographic maps is not genetically determined.
(b) Cortical topographic maps are fine-tuned by sensory experience.
(c) MEXICAN HAT refers to patterns of lateral connections in a neural network in which each node of the network has excitatory lateral connections with its closest neighbors, but inhibitory
lateral connections with more distant neighbors.
(d) If some regions of the input space are used more than others and receive a rich variety of stimulus patterns (e.g., hands of a pianist or a surgeon), Mexican hat will act to devote more cortical territory to processing inputs from those regions.
(e) When a part of a limb is amputated, the cortical regions that originally received input from this part of the body will gradually establish new afferent connections with neighboring surviving
parts of the limb.

11.  Which one of the following statements is WRONG?
(a) In the INVERTED MEXICAN HAT pattern of lateral connections, each node (locus) of the network has inhibitory lateral connections with its closest neighbors, but excitatory lateral
connections with more distant neighbors.
(b) Inverted Mexican hat causes RFs of cells in a local cortical region to become SHUFFLED; i.e., to stay in the same local region of the input space, but overlap only minimally among the
closest cells.
(c) Neurons within a minicolumn have very similar functional properties (very similar RFs), but adjacent minicolumns have dissimilar functional properties (their RFs overlap only minimally).
(d) The purpose of Inverted Mexican hat is to provide local cortical regions (groups of minicolumns) with diverse information about a local region of the input space.
(e) Overall, the pattern of lateral connections in the cortical input layer apparently is a combination of a small-scale Mexican hat and larger-scale Inverted Mexican hat.

12.  Which one of the following statements is WRONG?
(a) Velocity of a pendulum motion plotted against its position is an example of a TIME SERIES plot.
(b) A PHASE-SPACE PLOT can be constructed by plotting one variable against itself some fixed time later; e.g.,  Xt vs. Xt+Dt.
(c) A phase-space plot can be constructed by plotting a variable against its first derivative.
(d) A phase-space plot can be constructed by plotting two or more different variables against each other; e.g., Xt vs. Yt.
(e) Regardless of a particular approach to producing a phase-space plot, the resulting graphs will look qualitatively the same: they will all show either a single point, or a closed loop, or a
quasi-periodic figure, or a chaotic plot.

13.  Which one of the following statements is WRONG?
             A DYNAMICAL ATTRACTOR might be:
(a) FIXED POINT
(b) LIMIT CYCLE, or PERIODIC
(c) QUASI-PERIODIC
(d) CHAOTIC
(e) VERIDIC

14.  Which one of the following statements is WRONG?
(a) BIFURCATION PLOTS are used to show how the complexity of dynamics varies with a particular parameter of a neural network.
(b) As a general rule, if two different external stimuli to a neural network are very similar to each other, then their dynamical attractors will also be very similar to each other.
(c) CHAOS is a DETERMINISTIC DISORDER.
(d) Chaotic dynamics is distinguished by its SENSITIVITY TO INITIAL CONDITIONS; i.e., even very similar initial conditions will lead, over time, to very different outcomes.
(e) Each and every parameter of a neural network has an effect on the complexity of dynamics and the nature of its dynamical attractor.

15.  Which one of the following statements is WRONG?
(a) ERROR-CORRECTION LEARNING is a type of learning in neural networks where connection weights are adjusted as a function of error between the network’s desired and actual
outputs.
(b) ERROR-CORRECTION LEARNING is a type of UNSUPERVISED LEARNING.
(c) ERROR-BACKPROPAGATION LEARNING is a version of error-correction learning used in nonlinear multi-layered networks.
(d) ERROR-BACKPROPAGATION NETWORKS can vary in the number of input channels, number of layers of hidden units, and number of units in each hidden layer.
(e) ERROR-BACKPROPAGATION NETWORKS can, in principle, learn any transfer function, given enough hidden layers and enough units in those layers.

16.  Which one of the following statements is WRONG?
(a) Before attempting to learn environmental regularities, the cerebral cortex has to learn the identities of the environmental factors contributing to those regularities.
(b) Hidden environmental variables that are involved in regularities, but are reflected only implicitly in the known variables, can be learned from the known variables by looking for “suspicious
coincidences” among sensory events.
(c) Suspicious coincidences in environmental events can be detected by looking among various functions over various subsets of the known variables, searching for such pairs of functions over
non-overlapping subsets of variables that will show correlated behaviors.
(d) One of the principal functions of cortical spiny stellate cells is to discover causal factors operating in the outside world and learn inferential relations among them.
(e) Knowledge of the environmental regularities can be used to extract weak or noisy sensory signals or to infer object features (e.g., contours) not visible directly.

17.  Which one of the following statements is WRONG?
(a) Individual pyramidal cells – and more specifically, their principal dendrites – are the most likely candidates for performing the task of finding correlated functions over different sets of
environmental variables.
(b) Synaptic learning in dendrites is controlled both by presynaptic activity and by the output activity of the postsynaptic cell; the latter is signaled to each synapse by spikes that are
backpropagated from the soma up through each dendrite.
(c) Different principal dendrites of the same pyramidal cell receive the same sensory inputs, but learn to produce different, uncorrelated, outputs.
(d) The cortex captures inferential relations among environmental variables in its pattern of ascending, lateral and feedback connections.
(e) The basic mode of operation of the cerebral cortex is what might be called “filling-in missing parts of a picture.”

18.  Which one of the following statements is WRONG?
(a) The principal task of the CORTICAL OUTPUT LAYERS is to organize sensory inputs and deliver them to the basal dendrites of the pyramidal cells.
(b) BASAL DENDRITES OF PYRAMIDAL CELLS tune pyramidal cells to new orderly environmental features (i.e., inferentially useful environmental variables).
(c) LATERAL CONNECTIONS ON THE APICAL DENDRITES of pyramidal cells learn inferential relations among environmental variables.
(d) CORTICO-THALAMIC FEEDBACK disambiguates and fills in missing sensory information in the thalamus.
(e) Pyramidal cells in the HIGHER-ORDER CORTICAL AREAS tune to more complex orderly features of the environment and learn more complex inferential relations.

19.  Which one of the following statements is WRONG?
(a) In the visual system, sensory receptors are called PHOTORECEPTORS.
(b) There are 2 principal classes of photoreceptors in human retina: CONES (for color vision) and RODS (for grayscale vision).
(c) There are 3 types of cones in human retina, sensitive to different ranges of light wavelengths; one is sensitive to red, another to green, the third to blue colors.
(d) In the visual pathway from retina to the cerebral cortex, photoreceptors connect to BIPOLAR CELLS, which in turn connect to GANGLION CELLS.
(e) Ganglion cells are located in the thalamus.

20.  Which one of the following statements is WRONG?
(a) In the visual system, ganglion cells have circular RFs with center and opposite surround.
(b) The function of ganglion cells is to ENHANCE CONTRAST in the visual images projecting on the retina.
(c) Cells in layer 4 of the primary visual cortex, V1, have “SIMPLE” RFs – small RFs made up of a circular central region that is surrounded by a ring-like opposite-sign margin.
(d) A typical cell in layer 4 of V1 responds selectively to lines or edges of a particular orientation in a particular location in the visual field.
(e) Cells in the output layers of V1 have “COMPLEX” RFs – a cell will respond selectively to lines or edges of a very particular orientation but in a relatively wide range of locations in the visual field.

Correct answers:  1(c) 2(d) 3(a) 4(c) 5(a) 6(b) 7(b) 8(e) 9(d) 10(a) 11(e) 12(a) 13(e) 14(b) 15(b) 16(d) 17(c) 18(a) 19(e) 20(c)

PART 2  (25%)

1.  Draw a diagram of a MEXICAN HAT pattern of lateral connections:
2.  Draw a diagram of an INVERTED MEXICAN HAT pattern of lateral connections:
3.  Draw a diagram of the pattern of lateral connections in the cortical INPUT LAYER (i.e., a combination of Mexican Hat and Inverted Mexican Hat).  Briefly note the functions played by
different excitatory and inhibitory parts of this pattern.