1. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 1 Local Stability Analysis Step One: find stationary point(s) Step Two: linearize around all stationary points (using Taylor expansion), the Eigenvalues of the linearized problem determine nature of stationary point: Real parts: positive: growth of fluctuations, instability negative: decay of fluctuations, stability Imaginary parts: if present, solutions are oscillatory (spiraling) spiraling inward or outward if non-zero real parts Overall: point (asymptotically) stable if all real parts negative

2. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 2 Examples of nonlinear activation functions (transfer functions): a. b. c. c. rectified hyperbolic tangent b. “sigmoidal function” a. “half-wave rectification” Note: we will typically consider the activation function as a fixed property of our model neurons but real neurons can change their intrinsic properties.

3. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 3 The Naka-Rushton function P ½, the “semi-saturation”, is the stimulus contrast (intensity) that produces half of the maximum firing rate r max. N determines the slope of the non-linearity at P ½. A good fit for the steady state firing rate of neurons in several visual areas (LGN, V1, middle temporal) in response to a visual stimulus of contrast P is given by: Albrecht and Hamilton (1982)

4. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 4 Interaction of Excitatory and Inhibitory Neuronal Populations M EE vEvE M IE M EI Dale’s law: every neuron is either excitatory or inhibitory, never both Motivations: understand the emergence of oscillations in excitatory-inhibitory networks learn about local stability analysis Consider 2 populations of excitatory and inhibitory neurons with firing rates v: vIvI

5. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 5 Parameters: M EE = 1.25, M EI = -1, gamma E = -10Hz, tau E = 10ms M II = 0, M IE = 1, gamma I = 10 Hz, tau I = varying M EE vEvE vIvI M IE M EI [ ] + Mathematical formulation: Stationary point:

6. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 6 stationary point *nullclines, zero-isoclines * * Phase Portrait A: Stationary point is intersection of the nullclines. Arrows indicate direction of flow in different area of the phase space (state space). B: real and imaginary part of Eigenvalue as a function of tau I.

7. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 7 Linearization around stationary point gives the following matrix A with these Eigenvalues:

8. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 8 For tau I below critical value of 40ms, Eigenvalues have negative real parts: we see damped oscillations. Trajectory spirals to stable fixed point

9. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 9 When tau I grows beyond critical value of 40ms, a Hopf bifurcation occurs (here tau I= 50ms): stable fixed point → unstable fixed point + limit cycle Here, the amplitude of the oscillation grows until the non-linearity “clips” it.

10. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 10 Neural Oscillations interaction of excitatory and inhibitory neuron populations can lead to oscillations very important in, e.g. locomotion: rhythmic walking and swimming motions: Central Pattern Generators (CPGs) also very important in olfactory system (selective amplification) also oscillations in visual system: functional role hotly debated. Proposed as solution to binding problem: Idea: neural populations that represent features of the same object synchronize their firing

11. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 11 Binding Problem what and where (how) pathways in visual system how do you know what is where? circle triangle up down visual field neural representation Synchronization no yes spike trains

12. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 12 Competition and Decisions Motivation: ability to decide between alternatives is fundamental Idea: inhibitory interaction between neuronal populations representing different alternatives is plausible candidate mechanism The most simple system: Winner-take-all (WTA) network K1K1 K2K2

13. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 13 Stationary States and Stability The stationary states for K 1 =K 2 =120: e 1 = 50, e 2 = 0 e 2 = 50, e 1 = 0 e 1 = e 2 = 20 Linear stability analysis: 1) for e 1 = 50, e 2 = 0 : 2) for e 1 = e 2 = 20 : (τ=20ms) → “stable node” → “unstable saddle”

14. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 14 Matlab Simulation one unit wins the competition and completely suppresses the other Behavior for strong identical input: K 1 =K 2 =K=120

15. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 15 Continuous Neural Fields So far: individual units, with specific connectivity patterns Idea: abstract from individual neurons to continuous fields of neurons, where synaptic weights patterns become homogeneous interaction kernels Variant 1: continuous labeling of input or output domain Variant 2: continuous labeling of two- dimensional cortical space

16. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 16 Recurrent Simple Cell Model Question: how is orientation selectivity achieved? (feedforward vs. recurrent accounts)

17. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 17 Classic Hubel and Wiesel Model simple cell sums input from geniculate On and Off cells in particular constellation complex cell sums inputs from simple cells with same orientation but different phase preference

18. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 18 Recurrent Model Stimulus with orientation angle θ=0. A: amplitude, c: contrast, ε: small nonlinear amplification

19. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 19 Superior Colliculus and Saccades Representation of saccade motor command in superior colliculus: vector averaging Yarbus

20. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 20 A Simple Model of Saccade Target Selection Question: how do you select the target of your next saccade? Idea: competitive “blob” dynamics in 2 dimensional “neural field” layer of non-linear units with local excitation linear unit for global inhibition

21. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 21 Stability Analysis of Saccade Model Step 1: look for homogeneous stationary solutions Step 2: find range of β for which homogeneous stationary solution becomes unstable Step 3: simulate system (Matlab), observe behavior Step 4: estimate the size of the resulting blob as a function of β Reminder: Convolution

22. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 22 Example Run Initialization: 10 random spots of small activity, I=0, η small Gaussian iid noise time Result: a blob of activity forms at location determined by initial state and noise

23. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 23 Results of Analysis Step 1: look for homogeneous stationary solutions h 0 =0 works, β>1/A prevents fully active layer (A=area of layer) Step 2: find range of β for which homogeneous stationary solution becomes unstable for small local fluctuation from h 0 =0 to grow, we need β<1/2πσ 2 Step 3: simulate system (Matlab), observe behavior formation of single blob of activity suppressing all other activity in layer Step 4: estimate the size of the resulting blob as a function of β, σ

24. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 24 Matlab Code Fragments % initialize layer size = 50; h = zeros(size,size); for i=1:10 x = unidrnd(size); y = unidrnd(size); h(x,y)=h(x,y)+0.05; end % main loop while(1) active = (h>0); I = conv2(active, g, 'same') - beta*(sum(sum(active))); h = (1-alpha)*h + alpha*I + normrnd(0, noise, size, size); % display plots, etc. pause end % initialize layer size = 50; h = zeros(size,size); for i=1:10 x = unidrnd(size); y = unidrnd(size); h(x,y)=h(x,y)+0.05; end % main loop while(1) active = (h>0); I = conv2(active, g, 'same') - beta*(sum(sum(active))); h = (1-alpha)*h + alpha*I + normrnd(0, noise, size, size); % display plots, etc. pause end

25. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 25 Discussion of Saccade Model Positive: roughly consistent with anatomy/physiology explains how several close-by targets can win over strong but isolated target suggests why time to decision is longer in situations with several equally strong targets similar models used in modeling human performance in visual search tasks Limitations: only qualitative account in order to make precise quantitative predictions, it is typically necessary to take more physiological details into account, which are mostly unknown: exact connectivity patterns non-linearities more than one area is involved what are all the inputs?

26. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 26 Connection to Maximum Likelihood Estimation So far: purely bottom-up view: networks with this connectivity structure just happen to exhibit this behavior and this may be analogous to what the brain does New idea: use such dynamics to do Maximum Likelihood estimation Want: New idea: blob dynamics + vector decoding works better than doing direct vector decoding on the noisy inputs 1-d “blob” network with noisy input r: firing rate vector, Θ: stimulus parameter Population vector decoding: where c a is the preferred stimulus vector for unit a

27. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 27 Binocular Rivalry, Bistable Percepts Idea: extend WTA network by slow adaptation mechanism. Adaptation acts to increase semi- saturation of Naka Rushton non- linearity ambiguous figure binocular rivalry LR

28. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 28 Matlab Simulation β=1.5

29. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 29 Discussion of Rivalry Model Positive: roughly consistent with anatomy/physiology offers parsimonious mechanism for different perceptual switching phenomena, in a sense it “unifies” different phenomena by explaining them with the same mechanism Limitations: provides only qualitative account real switching behaviors are not so nice and regular and simple: cycles of different durations temporal asymmetries rivalry: competition likely takes place in hierarchical network rather than in just one stage. spatial dimension was ignored