Lecture 21 Neural Modeling II Martin Giese

Aim of this Class Account for experimentally observed effects in motion perception with the simple neuronal model presented in Lecture 18: Dependence on stimulus strength Stochastic nature of perception Psychometric function Bistable motion perception Hysteresis Perceptual switching

Model Neuron Population of real neurons  Similar tuning properties: –position –direction –speed Visual Field

Simplified Neuron Model Model for single local motion detector (consisting of many neurons with similar tuning) Average membrane potential (over all neurons) Stimulus intensity (input signal) Rate of change of average membrane potential “Time constant” Resting level parameter

Phase Diagram u u* u increasesu decreases attractor (u stays constant) u “Phase diagram”:

Attractor Follow the arrows in the phase diagram: u(t) changes always in the direction of the attractor u* u attractor u(t) t u* u(t) t u* u(t) t u*

With Noise State fluctuates near attractor u attractor u(t) t u* Probability u

With Noise (for Experts) Exact mathematical description: Stochastic Process (Ornstein-Uhlenbeck) u(t) t u* white noise

Single Neuron Model

Activation and Perception Assumption: Motion is seen only when u(t) is larger than a threshold value T. Stimulus seen No stimulus seen t u T u(t) t t T u

No Stimulus s = 0 u u* attractor “Phase diagram”: u T (threshold) -h motion seen

Large Stimulus s > 0 u u* attractor “Phase diagram”: T s-h motion seen u

Psychometric Function u T motion seen u u u P u P P u u Stimulus: no weak strong s P(seen) 1 0 Psychometric function: -h

Two Neuron Model Inhibition

Motivation Inhibition Bistable perception by lateral inhibition between motion detectors !

Two Uncoupled Neurons Neuron 1: Neuron 2: No inhibition.

2D Phase Diagram Attractor: pair of activations [u 1 *, u 2 *] u1 u1 u* u u2 u2 u attractor

Synaptic Coupling Neuron for “horizontal” Inhibition Neuron for “vertical” Coupling through nonlinear threshold function ! (Katz & Miledi, 1967)

Two Coupled Neurons Neuron 1: Neuron 2: Inhibition of neuron 1 by neuron 2 Inhibition of neuron 2 by neuron 1 f(u) u

2D Phase Diagram Two attractors !

2 Attractors (for Experts) Linear system theory predicts for purely linear interaction: not more than one isolated attractor if multiple attractors, they are not asymptotically stable (in all directions in phase space)

2 Attractors vertical motion seen Two attractors ! horizontal motion seen vertical motion not seen horizontal motion not seen

Bistability basin of attraction for vertical basin of attraction for horizontal  perceptual ambiguity

Perceptual Switching Fluctuations can push state in the other basin of attraction  perceptual switch

Influence of the Aspect Ratio Change of aspect ratio: different motion detectors activated change of the mutual inhibition InhibitionStimulus

Relative Stability  horizontal and vertical equally stable  horizontal more stable than vertical

Hysteresis State can not follow change of attractor immediately  tendency to stay in the same basin of attraction  hysteresis

Switching Probability  switch possible  switch very unlikely Medium aspect ratio Extreme aspect ratio

Programming

NeuDyn.m Simulates neural model with 1or 2 neurons and returns the time series of activation. ut = NeuDyn(tsim, u0, S, tau, Q, W);

NeuDyn.m Simulates neural model with 1or 2 neurons and returns the time series of activation. ut = NeuDyn(tsim, u0, S, tau, Q, W); simulation time steps time constant strength of fluctuations

NeuDyn.m Simulates neural model with 1or 2 neurons and returns the time series of activation. ut = NeuDyn(tsim, u0, S, tau, Q, W); initial activation stimulus strength (number / vector)

NeuDyn.m Simulates neural model with 1or 2 neurons and returns the time series of activation. ut = NeuDyn(tsim, u0, S, tau, Q, W); interaction (vector / matrix) simulated activity time series (vector / matrix)

NeuDyn.m Simulates neural model with 1or 2 neurons and returns the time series of activation. ut = NeuDyn(tsim, u0, S, tau, Q, W); simulation time steps initial activation time constant stimulus strength strength of fluctuations interaction simulated activity time series

Additional readings (for interested people): Hock, H.S., Kelso, J.A.S., Schöner, G. (1993). Bistability and hysteresis in the organization of apparent motion patterns. Journal of Experimental Psychology: Human Perception and Performance, 19(1), Giese, M.A. (1999). Dynamic Neural Field Theory of Motion Perception, Kluwer, Dordrecht, NL. Wilson, H.R. (1999). Spikes, Decisions, and Actions. Oxford University Press, Oxford, UK. Literature