1. Biological Modeling of Neural Networks: Week 11 – Continuum models: Cortical fields and perception Wulfram Gerstner EPFL, Lausanne, Switzerland 11.1 Transients - sharp or slow 11.2 Spatial continuum - model connectivity - cortical connectivity 11.3 Solution types - uniform solution - bump solution 11.4. Perception 11.5. Head direction cells Week 11 – Continuum models –Part 1: Transients

2. Single population full connectivity review from Week 10-part 3: mean-field arguments All neurons receive the same total input current (‘mean field’)

3. frequency (single neuron) Homogeneous network All neurons are identical, Single neuron rate = population rate A(t)= A 0 = const Single neuron Review from Week 10: stationary state/asynchronous activity

4. All spikes, all neurons fully connected All neurons receive the same total input current (‘mean field’) Index i disappears Week 10-part 3: mean-field arguments

5. h(t) I(t) ? A(t) potential A(t) t population activity Blackboard: h(t) Week 11-part 1: Transients in a population of uncoupled neurons Students: Which would you choose?

6. Connections 4000 external 4000 within excitatory 1000 within inhibitory Population - 50 000 neurons - 20 percent inhibitory - randomly connected -low rate -high rate input Week 11-part 1: Transients in a population of neurons

7. Population - 50 000 neurons - 20 percent inhibitory - randomly connected 100 200 time [ms] Neuron # 32374 50 u [mV] 100 0 10 A [Hz] Neuron # 32340 32440 100 200 time [ms] 50 -low rate -high rate input Week 11-part 1: Transients in a population of neurons

8. low noise I(t) h(t) noise-free (escape noise/fast noise) uncoupled population low noise fast transient I(t) h(t) (escape noise/fast noise) high noise slow transient But transient oscillations Week 11-part 1: Theory of transients for escape noise models

9. noise model A I(t) h(t) (escape noise/fast noise) high noise slow transient Population activity Membrane potential caused by input blackboard In the limit of high noise, Week 11-part 1: High-noise activity equation

10. noise model A I(t) h(t) (escape noise/fast noise) high noise slow transient Population activity Membrane potential caused by input 1 population = 1 differential equation Week 11-part 1: High-noise activity equation

11. Week 10-part 2: mean-field also works for random coupling full connectivityrandom: prob p fixed Image: Gerstner et al. Neuronal Dynamics (2014) random: number K of inputs fixed

12. Quiz 1, now Population equations [ ] A single cortical model population can exhibit transient oscillations [ ] Transients are always sharp [ ] Transients are always slow [ ] in a certain limit transients can be slow [ ] An escape noise model in the high-noise limit has transients which are always slow [ ] A single population described by a single first-order differential equation (no integrals/no delays) can exhibit transient oscillations

13. Biological Modeling of Neural Networks: Week 11 – Continuum models: Cortical fields and perception Wulfram Gerstner EPFL, Lausanne, Switzerland 11.1 Transients - sharp or slow 11.2 Spatial continuum - from multiple to continuous populations - cortical connectivity 11.3 Solution types - uniform solution - bump solution 11.4. Perception 11.5. Head direction cells Week 11 – part 2 :

14. Blackboard Week 11-part 2: multiple populations  continuum

15. Population activity Membrane potential caused by input 1 field = 1 integro-differential equation Week 11-part 2: Field equation (continuum model)

16. Exercise 1.1 now (stationary solution) Consider a continuum model, Find analytical solutions: - spatially uniform solution A(x,t)= A 0 Next lecture at 10:45 If done: start with Exercise 1.2 now (spatial stability)

17. Week 11-part 2: coupling across continuum xy Mexican hat

18. Week 11-part 2: cortical coupling

19. Biological Modeling of Neural Networks: Week 11 – Continuum models: Cortical fields and perception Wulfram Gerstner EPFL, Lausanne, Switzerland 11.1 Transients - sharp or slow 11.2 Spatial continuum - from multiple to continuous populations - cortical connectivity 11.3 Solution types - uniform solution - bump solution 11.4. Perception 11.5. Head direction cells Week 11 – part 3 :

20. Week 11-part 3: Solution types (ring model) Coupling: Input-driven regime Bump attractor regime

21. 0 A(x) I. Edge enhancement Weaker lateral connectivity I(x) Possible interpretation of visual cortex cells: (see final part this week) Field Equations: Wilson and Cowan, 1972 Week 11-part 3: Solution types: input driven regime

22. 0 A II: Bump formation: activity profile in the absence of input strong lateral connectivity Possible interpretation of head direction cells:  (see later today) Field Equations: Wilson and Cowan, 1972 Week 11-part 3: Solution types: bump solution

23. Exercise 2.1+2.2 now (stationary bump solution) Consider a continuum model, Find analytically the bump solutions w(x-y) Next lecture at 11:28

24. Continuum: stationary profile Comparison simulation of neurons and macroscopic field equation Spiridon&Gerstner See: Chapter 9, book: Spiking Neuron Models, W. Gerstner and W. Kistler, 2002 Week 11-part 3: Solution types: bump solution

25. Week 11-part 3: Solution types (continuum model) time

26. Biological Modeling of Neural Networks: Week 11 – Continuum models: Cortical fields and perception Wulfram Gerstner EPFL, Lausanne, Switzerland 11.1 Transients - sharp or slow 11.2 Spatial continuum - model connectivity - cortical connectivity 11.3 Solution types - uniform solution - bump solution 11.4. Perception 11.5. Head direction cells Week 11 – part 4:

27. Basic phenomenology 0 A(x) I. Edge enhancement Weaker lateral connectivity I(x) Possible interpretation of visual cortex cells: contrast enhancement in - orientation - location Field Equations: Wilson and Cowan, 1972 Week 11-part 5: uniform/input driven solution

28. Continuum models: grid illusion

29. Continuum models: Mach Bands

30. Week 11-part 4: Field models and Perception

33. Biological Modeling of Neural Networks: Week 11 – Continuum models: Cortical fields and perception Wulfram Gerstner EPFL, Lausanne, Switzerland 11.1 Transients - sharp or slow 11.2 Spatial continuum - model connectivity - cortical connectivity 11.3 Solution types - uniform solution - bump solution 11.4. Perception 11.5. Head direction cells Week 11 – part 5:

34. Basic phenomenology 0 A II: Bump formation strong lateral connectivity Possible interpretation of head direction cells: always some cells active  indicate current orientation Week 11-part 5: Bump solution

35. rat brain CA1 CA3 DG pyramidal cells soma axon dendrites synapses electrode Place fields Week 11-part 5: Hippocampal place cells

36. Main property: encoding the animal’s location place field Week 11-part 5: Hippocampal place cells

37. Main property: encoding the animal ’s heading   Preferred firing direction r (  i   i Week 11-part 5: Head direction cells

38. Main property: encoding the animal ’s allocentric heading Preferred firing direction r (  i   i 0 90 180 270 300 Week 11-part 5: Head direction cells

40. 11.1 Transients - sharp or slow 11.2 Spatial continuum - model connectivity - cortical connectivity 11.3 Solution types - uniform solution - bump solution 11.4. Perception 11.5. Head direction cells Week 11-Continuum models The END