version 0.10 (c) 2007 celest visi n brightness contrast: advanced modeling classroom presentation

Version 0.10 (c) 2007 CELEST

VISINBRIGHTNESS CONTRAST:

ADVANCED MODELINGCLASSROOM PRESENTATION


ANATOMY OF A NEURON


ANATOMY OF AN ACTION POTENTIAL

Neurons use action potentials to communicate with one another

An action potential occurs when an electrical charge travels down the axon from the cell body to the axon terminals

Axon

Axon Terminals

DendritesCell #1

Cell #2


HOW NEURONS COMMUNICATEAt the axon terminals the

electrical signal is converted to a chemical signal

These chemical signal are called neurotransmitters, which can be either excitatory or inhibitory

Neurotransmitters are released from the axon terminal through the synapse to the dendrite terminals of one or many other cells

Axon Terminal

Synapse

Neurotransmitter

Dendrite Terminal


A NEURON-INSPIRED MODEL

xi zij xj

vi eij vj

Source: http://webspace.ship.edu/cgboer/neuron.gif© Copyright 2003 C. George Boeree

xi Short-term memory traces

vi Cell populations

eij Axons

zij Long-term memory traces

xj Short-term memory traces

for the next neuron

vj Cell populations

Source: S. Grossberg (1988). Nonlinear neural networks: Principles, mechanisms, and architectures. Neural Networks, 1, 17-61.

Key:


GRAPHING CONVENTIONS

Modulators Learned weights

Excitation

Inhibition


TYPES OF CONNECTIONS

Convergent Divergent

“In-star” “Out-star”


TYPES OF CONNECTIONS

Feedforward Feedback


A MODEL OF BRIGHTNESS PERCEPTION


DIFFERENT TYPES OF RETINAL CELLS


Photoreceptors

Ganglion cells

+ +- --- --

A MASS ACTION MODEL

+Inhibitory Connections

Excitatory Connections


CENTER-SURROUND RECEPTIVE FIELD

The receptive field of a neuron is defined by the region of visual space where a stimulus will alter the firing rate of that neuron

Ganglion cells have a special receptive field called center-surround because of competitive interaction


COMPETITIVE INTERACTION

Inhibition

Excitation

+-

-

-

-+

+

+

Stimulus On

Stimulus On

Stimulus Off

Stimulus Off

Firing Rate

Firing Rate


LATERAL INHIBITON

In diffuse light conditions, light hits both the On-center and Off-surround, providing about equal level of excitation and inhibition to the bipolar cell, giving a baseline firing rate

When light hits the photoreceptor in the On-center only, it sends a signal through the bipolar cell, and the ganglion cell is excited above baseline

When light excites rods/cones in only the Off-surround, causing the horizontal cells to send inhibitory signals through the bipolar cell to the ganglion cell which is suppressed below baseline. This is called lateral inhibition


MACH BAND ILLUSION

+-

-

-+

-

-

-

+-

-

-+

-

-

-

1.

2.

3.

4.

Graph of Perceived Brightness


+ + +- --- --

+ + +

I1 I2 I3

1

3

2

x1 x2 x3

MODEL LAYER

Visual Light Input (Ii)

Photoreceptors (I)

Ganglion Cells (xi)

Inhibitory Indirect

Pathway (-)

Excitatory Direct Pathway (+)


INPUT-BASED EXCITATION: AN ACTION POTENTIAL

Our independent variable is the change of the ganglion cell membrane potential over time:

dxi /dt

Our dependent variable is visual input: I

So fundamentally our equation is:

dxi /dt = I

Input-based excitation (dxi /dt)

Visual Input (I)


SPONTANEOUS DECAY

Neurons that are not being continuously excited quickly return to resting

potential

To model this, we add a decay term -Axi, so the neuron will return to its resting potential at a rate proportional to its level of excitation:

dxi /dt = -Axi + I

Passive decay of

activation


EXCITING A POST-SYNAPTIC NEURON

The level of excitation a neuron can receive is a function of how many synaptic connections a neuron’s dendrite has, as well as how many receptor sites there are per synapse

A constant parameter, B, will be used to represent the maximum excitation that a neuron can receive


EXCITATION HAS A LIMIT

The capacity of unused excitatory sites is represented by B-xi. The total rate at which a cell’s level of excitation can increase is (B-xi)I

This has two effects:

1. The value of xi must be less than or equal to B

2. If an unexcited cell and an excited cell receive the same size inputs (I) the unexcited cell will have a larger increase in activity than the excited one. We can now update the equation to:

dxi /dt = -Axi + (B-xi)I


COMPETITIVE INHIBITION

Next, we need to subtract the neighboring connections because they produce lateral inhibition.

We can represent the inhibitory connections with:

Ii - ∑ (k≠i)Ik

Where: I = total visual field

Ii = excitatory input

Ik = inhibitory input

This updates our model to:

dxi /dt = -Axi + (B-xi)Ii - ∑ (k≠i)Ik

+ + +- --- --

+ + +

I1 I2 I3

1

3

2

x1 x2 x3


INHIBITION HAS A LIMIT

Just like the excitatory sites, there is a limited number of potential inhibitory connection sites. We will set the number of possible inhibitory sites to C

We will represent the inactive inhibitory sites as -xi - C or -(xi + C), and the total rate at which inhibition can increase as -(xi + C)Ik

If the inputs Ik are greater than Ii, xi will decrease to –C. However, if Ii ,is greater than the other inputs xi will increase to B. This produces the final form of the model:

dxi /dt = -Axi + (B-xi)Ii - (xi+C) ∑(k≠i)Ik


EQUATION REVIEW

Property Equation

Input Excitation dxi /dt = Ii

Spontaneous Decay dxi /dt = -Axi + Ii

Limited Excitation dxi /dt = -Axi + (B-xi)Ii

Competitive Inhibition

dxi /dt = -Axi + (B-xi)Ii - ∑(k≠i) Ik

Limited Inhibition dxi /dt = -Axi + (B-xi)Ii -(xi + C) ∑(k≠i) Ik


PARAMETER REVIEWParameter Definition

xiGanglion cell response

dt Time delay between each incremental time step

dxi /dt Rate of change of the ganglion cell response (xi) for each time step (dt)

iPosition of cells at each step being measured

kPosition of every other cell at each step NOT being measured

A Decay rate. The larger the decay rate, the faster the ganglion cells will return to resting potential (0)

B Upper limit any ganglion cell response can reach

-C Lower limit that any given ganglion cell response can reach. The lower the ganglion cell response can go, the harder it will be for the ganglion cell to reach threshold


= -Axi+(B-xi)Ii-(xi+C)∑(k≠i) Ikdxi

dt

Rate of Change of Ganglion Cell Response =

Spontaneous Decay + Excitation – Inhibition

ABSTRACT MATHEMATICAL MODEL REVIEW


MODEL SOFTWARE

version 0.10 (c) 2007 celest visi n brightness contrast: advanced modeling classroom presentation

Documents

neuron slide

celest anatomy

star slide

lateral inhibition slide

firing rate slide

competitive interaction

bipolar cell

celest lateral inhibiton