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 17:30 – 18:45, Communication Bldg. 115.

Prerequisites:  Good computer programming skills.

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:  Monday, Wednesday 12 - 2 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
        - The spiked random neural network (RNN)
        - Synaptic plasticity: SINBAD hierarchical networks
        - Self-organization of cortical networks: Neural Mechanics

    Local network dynamics
        - Functional architecture of CNS networks (emphasis on cerebral cortex)
        - Attractor dynamics
        - Nonlinear dynamical concepts in neural systems
        - Cerebral cortical dynamics

    Information processing in cortical output layers
        - SINBAD mechanism for discovery of key hidden environmental variables
        - Building an internal working 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

    Computer modeling projects (throughout the course)
        - RNN based pattern recognition algorithm
        - Point electrical model of a neuron
        - SINBAD network
        - Neural Mechanics

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 “resting” channels.

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), 115-116 (ion channels)

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.
        E_Na+ , equilibrium potential of Na+ channels is +40 mV.
        E_K+ , equilibrium potential of K+ channels is -90 mV.
        E_Cl-, 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 (AP) 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).

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.

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 g_m, in series with battery whose E = -70 mV, or resting membrane potential.
g_m 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 w_i * A_i,   where w_i (or g_i) is the weight of synapse i and A_i is the activity of the presynaptic cell i.

An entire neuron can be represented by an electrical circuit that consists of a number of components all connected in parallel:
- capacitor C_m, representing membrane capacitance
- resistor g_m in series with a battery E_r = -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 Gin_K in series with a battery Ein_K = -90 mV, representing net conductance of all the inhibitory synapses that use “subtracting” K+ ion channels
- variable resistor Gin_Cl in series with a battery Ein_Cl = -70 mV, representing net conductance of all the inhibitory synapses that use “dividing” Cl- ion channels
- variable resistor gAP_Na in series with a battery E_Na= +40 mV, representing voltage-gated Na+ channels responsible for generation of action potential
- variable resistor gAP_K in series with a battery E_K = -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:

                          g_m* Er + Gex * Eex + Gin_K* Ein_K + Gin_Cl* Ein_Cl + gAP_Na* E_Na + gAP_K* E_K
        V  =     --------------------------------------------------------------------------------------
                                              g_m + Gex + Gin_K + Gin_Cl+ gAP_Na + gAP_K
 

Text:   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).

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
                        g_m = 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  :
                                    Gex_t = (1-1/t) * Gex_t-1  +  (1/t) * Cex *  S(w⁺_i  *  A⁺i _t )

                                    Gin_t = (1-1/t) * Gin_t-1  +  (1/t) * Cin * S(w^-_i   *  A^-i_t )

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

            DV = ( 70 * Gex_t ) / ( Gex_t +  Gin_t  + g_m )

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., 50) different values of Cex.
In each run, do 20 time steps and save the value of  DV  on the last, 20th time step (by then DV 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:   February 6.

                                                         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 = 100   - number of time steps (in Exercises 2 and 3, Ntimes = 20)
                            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).

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. 10 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:

                                    GAMPA_t = (1-1/t_AMPA) * GAMPA_t-1  +  (1/t_AMPA) * C_AMPA *  S(w⁺_i  *  A⁺i _t )

                                    GNMDA_t = (1-1/t_NMDA) * GNMDA_t-1  +  (1/t_NMDA) * C_NMDA *   S[(w⁺_i  *  A⁺i _t ) * DV]+

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

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

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).
 

NEUROMODULATION

Neuromodulators are chemical compounds, released from synapses of special neurons, that temporarily modify properties of receptors/channels.
4 basic neuromodulators: acetylcholine, norepinephrine, serotonin, dopamine

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

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, V_i = 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.

HOMEWORK ASSIGNMENT #2

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
          P channels                             T

P channels are special channels, permeable to Na+ and voltage-gated (more depolarized the local membrane, more channels open).
P stands for "persistent" - as long as V is high, the channels stay open.
As a part of your homework, suggest what might be the purpose of these P channels in the terminal tuft.

Due date:  February 12.  Submit a drawing of the electrical circuit representing this cell and suggestion for P channel function.

REQUIRED Reading:  C. Koch (1997)  Computation and the single neuron. Nature 385: 207-210
This article has the answer to the P channel question.
 

HOMEWORK ASSIGNMENT #3

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

        channel type:      GABA_A             GABA_B
        transmitter:         GABA                GABA
        receptor:             GABA_A            GABA_B
        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 (GABA_A and GABA_B channels)
(3)  inhibitory (GABA_A 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:  February 18

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

Text:  pp. 10 –11, 77 - 88

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

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).

Text:  pp. 86-87, 324-329, 369-375.

Modeling Project:  Receptive Field of a Neuron.

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:
        Brief 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

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. 99-104.

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 w_max.

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 -
           Dw_i = (As_i – As_i) * (At – At) * RL
        Next compute tentative new weights -  w’_i = w_i + Dw_i
        If the sum of all w’_i (Sw’_i) is less or equal w_max, then new weights are w_i = w’_i
        If  Sw’_i > w_max, then new weights are w_i = w’_i* w_max / Sw’_i

This way, sum of weights will never exceed wmax.

Text:  pp. 680-681.

If a target cell receives synaptic connections from a number of source cells and these connections can change their weights by some version of Hebb Rule, then - depending on the particular mathematical version of Hebb Rule - the target cell can learn to extract different information from its inputs.

(1)  Oja's version of Hebb Rule

At a given time moment the change in the weight of connection from source cell i is:
 Dw_i = As_i * At * RL - At² * w_i
            where w_i  is the current weight of the connection,
            As_i is the activity of the source cell i (e.g., instantaneous firing rate),
           At is the output activity of the target cell and
           LR is “rate of learning” scaling constant.

Under this learning rule, the target cell will learn to compute and represent by its activity the First Principal Component of its input. That is, it will perform "PCA" (Principal Component Analysis, which is a popular technique in data compression). The First Principal Component is the direction in the input space (defined by activities of all the source cells) that has the maximal variance.  Thus, PCA-performing Hebb Rule can be used in groups of neurons to compress information coming to them from other brain regions, so that the same information will be represented by fewer neurons.

(2)  BCM version of Hebb Rule

Dw_i = As_i * At * (At - At²) *RL
            where At²  is the average of the squared activity of the target cell.

This version of Hebb Rule performs "ICA" (Independent Component Analysis).  That is, suppose the set of source cells carry information about several important sensory variables x₁ ... x_n but, unfortunately, these variables are mixed together:  activity of each source cell is a linear sum of the variables -   As_i = Sw_ij * x_j
How can we extract x₁ ... x_n from As₁  ... As_m ?  They can be extracted by ICA (see http://www.cs.ucf.edu/~kursun/fav_ica.doc)
Here x₁ ... x_n  are called "Independent Components".  BCM version of Hebb Rule will make the target cell to learn to represent one of the variables x₁ ... x_n.  If we have several target cells, then together they can learn to represent by their activities all the hidden variables x₁ ... x_n.

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.

HOMEWORK ASSIGNMENT #4

MODELING PROJECT:  AFFERENT GROUP SELECTION

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 g_m = 1 (E = -70) and net excitatory conductance Gex (E = 0).
Activity of the target cell is:   DV = 70 * Gex / (Gex + g_m)

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 w_i 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)
                        - 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

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 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.

Optional reading: Favorov and Kelly (1994) Network's afferent connectivity is governed by neural mechanics. Society for Neuroscience Abstracts.

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.

General layout of topographic maps is genetically determined (see lecture #11), 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
              Florence and Kaas (1995) Somatotopy: plasticity of sensory maps. In: The Handbook of Brain Theory and Neural Networks, M.A. Arbib, (ed), Bradford Books/MIT Press, pp. 888-891

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.

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 14).

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.

Take-home test

Read the following three papers:

Shadlen and Newsome (1994) Noise, neural codes and cortical organization. Current Opinion in Neurobiology 4: 569-579
Softky (1995) Simple codes vsersus efficient codes. Current Opinion in Neurobiology 5: 239-247
Shadlen and Newsome (1995) Is there a signal in the noise?. Current Opinion in Neurobiology 5: 248-250

Briefly summarize the question addressed by these papers, the alternative answers offered by the opponents, and the essence of their arguments in favor of their positions.

Due:  April 8.

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)

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.

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.

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.

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.
Reading:  Eberhart (1989)  Chaos theory for the biomedical engineer. IEEE Engineering in Medicine and Biology 8: 41-45
               Favorov et al. (2002) Spurious dynamics in somatosensory cortex
Optional:  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

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 #5:  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(Win_1i * IN1 + Win_2i * IN2)
tanh() is the hyperbolic tangent function y =(e^x-e^-x)/(e^x+e^-x).

Activity of the output unit is:  OUT = tanh( S Wh_i * 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:   Wh_i = Wh_i + d * Hi * RLh
                      where RLh = 0.003 is rate of learning
      (4)  Backpropagate error signal to each hidden unit:  d_i = d * Wh_i * (1- Hi²)
       (5)  Adjust input unit connections:   Win_ij = Win_ij + d_j * 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
 

Optional Reading:   J. Anderson (1995)  An Introduction to Neural Networks.  ISBN 0-262-01144-1, chapter 9, pp. 239-280.

Lectures 25-26, April 15-17:  SINBAD MODEL OF CORTICAL ORGANIZATION AND FUNCTION

MAIN READING:
     Download from www.sinbad.info: Favorov and Ryder. SINBAD: a neocortical mechanism for discovering environmental variables and regularities hidden in sensory input.

Also Required Reading:
      Clark and Thornton (1997)  Trading spaces: computation, representation, and the limits of uninformed learning.  Behavioral and Brain Sciences 20: 57-66.
      Perlovsky (1998) Conundrum of combinatorial complexity. IEEE Trans. on Pattern Analysis and Machine Intelligence 20: 666-670.
 

FUNCTIONAL ORGANIZATION OF CEREBRAL CORTEX (OVERVIEW)

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

1.  Describe how neurons communicate with each other (synaptic transmission).

2.  Describe major subdivisions of the nervous system, partitioning of the cerebral cortex, and general organization of sensory pathways to the cerebral cortex.  Indicate functional roles of the identified brain regions.

3.  Give concise overview of the topographic organization of cerebral cortical areas: define the concept of a Receptive Field, describe basic properties of a typical topographic map, list (but do not explain how they work) the major mechanisms of topographic map formation and contributions of those mechanisms to cortical topography.

4.  Describe Hebbian synaptic plasticity, and its consequences for the development of patterns of afferent connections to cortical neurons. Include discussion of Mexican Hat and Inverted Mexican Hat patterns of lateral connections, how they affect the afferent connections to cortical neurons, and the roles they play in formation of cortical topographic maps.

5.  Describe cortical functional architecture, including laminar subdivisions, flow of information across layers, connectional organization of the output layers (lateral connections), and their interpretation in the context of the SINBAD model of the cortical network.

6.  Describe nonlinear dynamical properties of the cortical network:  define phase-space plots, bifurcation plots, concept of a dynamical attractor, and types of attractors.  What does distinguish chaotic dynamics? What kinds of dynamics are present in the cortical network?

7. Discuss the importance of knowing the hidden, but influential environmental variables. How can such hidden variables be discovered? Outline the SINBAD model of the cortical pyramidal cell. How can a network of SINBAD cells function as an INFERENTIAL model of the outside world? How can such an internal model be used to generate successful behaviors?
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During the exam, you will be given two of these questions. You will have to write your answers without any help from notes, books, etc. The third question will be either (1) to draw a compartmental model of a particular neuron, or (2) to write equations describing a point model of a particular neuron.

If you have any questions, you can see me any afternoon before the exam.

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NEUROSCIENTIFIC TERMS THAT YOU NEED TO KNOW:
 (This list of terms will be provided to you during the final exam)

Neuron   Dendrites    Soma    Axon hillock    Axon    Axon terminals    Synapse
Action potential    Spike
Neuronal membrane     Ion pump       Ion channel     Na+/K+ ion pump
K+  Na+  and  Cl-  resting ion channels
GATED ION CHANNELS
Concentration gradient   Electrical gradient
MEMBRANE POTENTIAL    RESTING POTENTIAL
HYPERPOLARIZATION     DEPOLARIZATION     Excitation    Inhibition
EQUILIBRIUM POTENTIAL
        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.
Action potential generating voltage-gated Na+ channels
Action potential generating voltage-gated K+ channels
Action potential threshold = -55 mV
ABSOLUTE REFRACTORY PERIOD        RELATIVE REFRACTORY PERIOD
SYNAPTIC TRANSMISSION      PRESYNAPTIC NEURON       POSTSYNAPTIC NEURON
Synaptic cleft    Synaptic vesicles
TRANSMITTER      RECEPTORS     RECEPTOR-GATED ION CHANNELS
POSTSYNAPTIC POTENTIAL (PSP)     EXCITATORY PSP (EPSP)      INHIBITORY PSP (IPSP)
Inhibitory synapses      Excitatory synapses
SYNAPTIC INTEGRATION

SPATIAL SUMMATION      TEMPORAL SUMMATION

SYNAPTIC EFFICACY       Connection “weight” (or strength)
Passive membrane conductance      Membrane capacitance
Channel conductance in series with a battery
POINT NEURON MODEL       COMPARTMENTAL MODEL
ADAPTATION     Presynaptic adaptation     Postsynaptic receptor/channel desensitization       AFTERHYPERPOLARIZATION

NERVOUS SYSTEM
PERIPHERAL NERVOUS SYSTEM      CENTRAL NERVOUS SYSTEM (CNS)
 SENSORY NERVES      MOTOR NERVES
BRAIN     SPINAL CORD
BRAINSTEM     CEREBELLUM      DIENCEPHALON
CEREBRAL HEMISPHERES
CEREBRAL CORTEX      LOBES
FRONTAL LOBE     PARIETAL LOBE     OCCIPITAL LOBE     TEMPORAL LOBE      INSULA      LIMBIC LOBE
CORTICAL AREAS     PRIMARY CORTICAL AREAS     SECONDARY CORTICAL AREAS        ASSOCIATIVE CORTICAL AREAS

SENSORY PATHWAYS
SENSORY RECEPTORS      PRIMARY AFFERENT NEURONS
SENSORY NUCLEUS        RELAY NUCLEUS in THALAMUS
MOTOR CONTROL SYSTEM
PREFRONTAL CORTEX       PREMOTOR CORTEX
PRIMARY MOTOR CORTICAL AREA
MOTOR NUCLEI        SKELETAL MUSCLES
BASAL GANGLIA
MOTIVATIONAL SYSTEM       HYPOTHALAMUS           AMYGDALA          LIMBIC CORTEX           RETICULAR FORMATION
CORTICAL TOPOGRAPHY
 POINT-TO-POINT        SMALL AREA-TO-SMALL AREA
TOPOGRAPHIC MAPS

RECEPTIVE FIELD (RF)         RECEPTOR SURFACE
Genetic mechanism of generating topographic maps in cortical areas
SYNAPTIC PLASTICITY
HEBBIAN SYNAPTIC PLASTICITY         HEBBIAN RULE
LONG-TERM POTENTIATION (LTP)          LONG-TERM DEPRESSION (LTD)
AFFERENT GROUP
LATERAL EXCITATION = ATTRACTION of afferent groups
LATERAL INHIBITION = REPULSION of the afferent groups
MEXICAN HAT        INVERTED MEXICAN HAT
Cortical layers      UPPER LAYERS       MIDDLE LAYERS          DEEP LAYERS
INPUT LAYER         OUTPUT LAYERS
COLUMNAR ORGANIZATION        CORTICAL COLUMNS          MINICOLUMNS
Locally shuffled receptive fields

 PYRAMIDAL cells             SPINY STELLATE cells

 LONG-RANGE HORIZONTAL CONNECTIONS

NONLINEAR DYNAMICAL SYSTEM     CHAOTIC DYNAMICS   EMERGENT BEHAVIORS
TIME SERIES     PHASE-SPACE PLOTS         BIFURCATION PLOTS
STEADY-STATE DYNAMICS     TRANSIENT DYNAMICS
DYNAMICAL ATTRACTOR
FIXED POINT     LIMIT CYCLE     PERIODIC
QUASI-PERIODIC    CHAOTIC
 DETERMINISTIC DISORDER       SENSITIVITY TO INITIAL CONDITIONS
TRANSIENTS

INPUT-TO-OUTPUT TRANFER FUNCTIONS.
ERROR-CORRECTION LEARNING    SUPERVISED LEARNING.
ERROR-BACKPROPAGATION LEARNING      BACKPROP NETS

BASAL DENDRITES OF PYRAMIDAL CELLS
APICAL DENDRITES
CORTICO-THALAMIC FEEDBACK
HIGHER-ORDER CORTICAL AREAS
CORTICO-CORTICAL FEEDBACK