Version 0.10 (c) 2007 CELEST VISI N BRIGHTNESS CONTRAST: ADVANCED MODELING CLASSROOM PRESENTATION.
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Transcript of 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
Version 0.10 (c) 2007 CELEST
ANATOMY OF A NEURON
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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
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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
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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:
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GRAPHING CONVENTIONS
Modulators Learned weights
Excitation
Inhibition
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TYPES OF CONNECTIONS
Convergent Divergent
“In-star” “Out-star”
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TYPES OF CONNECTIONS
Feedforward Feedback
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A MODEL OF BRIGHTNESS PERCEPTION
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DIFFERENT TYPES OF RETINAL CELLS
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Photoreceptors
Ganglion cells
+ +- --- --
A MASS ACTION MODEL
+Inhibitory Connections
Excitatory Connections
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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
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COMPETITIVE INTERACTION
Inhibition
Excitation
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Stimulus On
Stimulus On
Stimulus Off
Stimulus Off
Firing Rate
Firing Rate
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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
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MACH BAND ILLUSION
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1.
2.
3.
4.
Graph of Perceived Brightness
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+ + +- --- --
+ + +
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 (+)
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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)
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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
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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
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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
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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
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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
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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
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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
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= -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
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MODEL SOFTWARE