1. Version 0.10 (c) 2007 CELEST VISI  N BRIGHTNESS CONTRAST: ADVANCED MODELING CLASSROOM PRESENTATION

2. Version 0.10 (c) 2007 CELEST ANATOMY OF A NEURON

3. Version 0.10 (c) 2007 CELEST 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 Dendrites Cell #1 Cell #2

4. Version 0.10 (c) 2007 CELEST HOW NEURONS COMMUNICATE At 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

5. Version 0.10 (c) 2007 CELEST A NEURON-INSPIRED MODEL x i z ij x j v i e ij v j Source: http://webspace.ship.edu/cgboer/neuron.gif © Copyright 2003 C. George Boeree xixi Short-term memory traces vivi Cell populations e ij Axons z ij Long-term memory traces xjxj Short-term memory traces for the next neuron vjvj Cell populations Source: S. Grossberg (1988). Nonlinear neural networks: Principles, mechanisms, and architectures. Neural Networks, 1, 17-61. Key:

6. Version 0.10 (c) 2007 CELEST GRAPHING CONVENTIONS ModulatorsLearned weights Excitation Inhibition

7. Version 0.10 (c) 2007 CELEST TYPES OF CONNECTIONS ConvergentDivergent “In-star” “Out-star”

8. Version 0.10 (c) 2007 CELEST TYPES OF CONNECTIONS Feedforward Feedback

9. Version 0.10 (c) 2007 CELEST A MODEL OF BRIGHTNESS PERCEPTION

10. Version 0.10 (c) 2007 CELEST DIFFERENT TYPES OF RETINAL CELLS

11. Version 0.10 (c) 2007 CELEST Photoreceptors Ganglion cells ++ - -- -- - A MASS ACTION MODEL + Inhibitory Connections Excitatory Connections

12. Version 0.10 (c) 2007 CELEST 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

13. Version 0.10 (c) 2007 CELEST COMPETITIVE INTERACTION Inhibition Excitation + - - - - + + + Stimulus On Stimulus Off Firing Rate

14. Version 0.10 (c) 2007 CELEST 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

15. Version 0.10 (c) 2007 CELEST MACH BAND ILLUSION + - - - + - - - + - - - + - - - 1. 2. 3. 4. Graph of Perceived Brightness

16. Version 0.10 (c) 2007 CELEST + + + - -- - - - + + + I1I1 I2I2 I3I3 132 x1x1 x2x2 x3x3 MODEL LAYER Visual Light Input (I i ) Photoreceptors (  I ) Ganglion Cells (x i ) Inhibitory Indirect Pathway (-) Excitatory Direct Pathway (+)

17. Version 0.10 (c) 2007 CELEST INPUT-BASED EXCITATION: AN ACTION POTENTIAL Our independent variable is the change of the ganglion cell membrane potential over time: dx i /dt Our dependent variable is visual input: I So fundamentally our equation is: dx i /dt = I Input-based excitation ( dx i /dt ) Visual Input ( I )

18. Version 0.10 (c) 2007 CELEST SPONTANEOUS DECAY Neurons that are not being continuously excited quickly return to resting potential To model this, we add a decay term -Ax i, so the neuron will return to its resting potential at a rate proportional to its level of excitation: dx i /dt = -Ax i + I Passive decay of activation

19. Version 0.10 (c) 2007 CELEST 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

20. Version 0.10 (c) 2007 CELEST EXCITATION HAS A LIMIT The capacity of unused excitatory sites is represented by B-x i. The total rate at which a cell’s level of excitation can increase is (B-x i )I This has two effects: 1. The value of x i 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: dx i /dt = -Ax i + (B-x i )I

21. Version 0.10 (c) 2007 CELEST COMPETITIVE INHIBITION Next, we need to subtract the neighboring connections because they produce lateral inhibition. We can represent the inhibitory connections with: I i - ∑ (k≠i) I k Where: I = total visual field I i = excitatory input I k = inhibitory input This updates our model to: dx i /dt = -Ax i + (B-x i )I i - ∑ (k≠i) I k + + + - -- - - - + + + I1I1 I2I2 I3I3 132 x1x1 x2x2 x3x3

22. Version 0.10 (c) 2007 CELEST 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 -x i - C or -(x i + C), and the total rate at which inhibition can increase as -(x i + C)I k If the inputs I k are greater than I i, x i will decrease to –C. However, if I i,is greater than the other inputs x i will increase to B. This produces the final form of the model: dx i /dt = -Ax i + (B-x i )I i - (x i +C) ∑ (k≠i) I k

23. Version 0.10 (c) 2007 CELEST EQUATION REVIEW PropertyEquation Input Excitation dx i /dt = I i Spontaneous Decay dx i /dt = -Ax i + I i Limited Excitation dx i /dt = -Ax i + (B-x i )I i Competitive Inhibition dx i /dt = -Ax i + (B-x i )I i - ∑ (k≠i) I k Limited Inhibition dx i /dt = -Ax i + (B-x i )I i -(x i + C) ∑ (k≠i) I k

24. Version 0.10 (c) 2007 CELEST PARAMETER REVIEW ParameterDefinition xixi Ganglion cell response dt Time delay between each incremental time step dx i /dt Rate of change of the ganglion cell response (x i ) for each time step ( dt) i Position of cells at each step being measured k Position 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

25. Version 0.10 (c) 2007 CELEST = -Ax i +(B-x i )I i -(x i +C)∑ (k≠i) I k dx i dt Rate of Change of Ganglion Cell Response = Spontaneous Decay + Excitation – Inhibition ABSTRACT MATHEMATICAL MODEL REVIEW

26. Version 0.10 (c) 2007 CELEST MODEL SOFTWARE