Connected Populations: oscillations, competition and spatial continuum

1. Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram Gerstner, EPFL Book: W. Gerstner and W. Kistler, Spiking Neuron Models Chapter 9.1

2. t h(t) potential A(t) A(t) population activity A(t) A(t) Population of neurons Blackboard: Pop. activity ? I(t)

3. Population - 50 000 neurons - 20 percent inhibitory - randomly connected Connections 4000 external 4000 within excitatory 1000 within inhibitory -low rate -high rate input Signal transmission in populations of neurons

4. -low rate -high rate input Signal transmission in populations of neurons A [Hz] 10 32440 Neuron # 32340 time [ms] 200 50 100 100 Neuron # 32374 u [mV] Population - 50 000 neurons - 20 percent inhibitory - randomly connected 0 time [ms] 200 50 100

5. Theory of transients noise model A (escape noise/fast noise) high noise I(t) I(t) h(t) h(t) But transient oscillations noise model A (escape noise/fast noise) low noise low noise noise-free fast transient slow transient

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

7. Part I: Single Population - Population Activity, definition - high noise - full connectivity Full connectivity

8. fully connected N >> 1 Fully connected network blackboard All spikes, all neurons Synaptic coupling

9. fully connected Index i disappears All neurons receive the same total input current (‘mean field’) All spikes, all neurons

10. fully connected N >> 1 Stationary solution Homogeneous network, stationary, All neurons are identical, Single neuron rate = population rate

11. fully connected N >> 1 Exercises 1, now Next lecture 10:15 Exercise 1: homogeneous stationary solution Homogeneous network All neurons are identical, Single neuron rate = population rate

12. Connected Populations: Part I: Single Population - Population Activity, definition - high noise - full connectivity - stationary state - mean rate in stationary state What is this function g? Wulfram Gerstner, EPFL

13. Stationary State/Asynchronous State input potential fullyconnected coupling J/N A(t)= A0= const Homogeneous network All neurons are identical, Single neuron rate = population rate Single neuron blackboard frequency (single neuron)

14. noise model A (escape noise/fast noise) high noise I(t) h(t) Note: total membrane potential Effect of last spike High-noise activity equation What is this function g? Population activity Membrane potential caused by input slow transient

15. j Spike reception: EPSP Spike reception: EPSP Firing: Spike Response Model Spike emission i Spike emission: AP All spikes, all neurons Last spike of i hi(t)

16. fully connected N >> 1 Fully connected network Spike emission: AP All spikes, all neurons Synaptic coupling

17. Stationary State/Asynchronous State A(t)=const input potential fullyconnected coupling J/N typical mean field (Curie Weiss) Homogeneous network All neurons are identical, Single neuron rate = population rate frequency (single neuron)

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

19. Wulfram Gerstner EPFL Connected Populations: Part I: Single Population - Population Activity, definition - high noise - full connectivity - stationary state - mean rate in stationary state Part II: Multiple Population - Spatial coupling - Background: cortical populations - Continuum model - Stationary state

20. Microscopic vs. Macroscopic Coupling I(t)

21. Cortical columns: Orientation tuning (and coarse coding)

22. electrode Detour: Receptive fields (see also lecture 4) visual cortex

23. Detour: Receptive field development visual cortex

24. Detour: Receptive field development Receptive fields: Retina, LGN + - -

25. Detour: Receptive field development Receptive fields: Retina, LGN Receptive fields: visual cortex V1 Orientation selective

26. Detour: Receptive field development Receptive fields: visual cortex V1 Orientation selective

27. Detour: orientation selective receptive fields Receptive fields: visual cortex V1 rate 0 Orientation selective Stimulus orientation

28. Detour: Receptive fields visual cortex Neighboring cells in visual cortex Have similar preferred orientation: cortical orientation map

29. Detour: orientation selective receptive fields Receptive fields: visual cortex V1 rate Cell 1 0 Orientation selective Stimulus orientation

30. Oriented stimulus Detour: orientation selective receptive fields rate Cell 1 Cell 5 Course coding Many cells respond to a single stimulus with different rate 0

31. Continuous Networks Continuous Networks

32. Continuum Several populations Blackboard

33. Field equations and continuum Models for coupled populations Part I: Single Population - Population Activity, definition - high noise - full connectivity - stationary state - mean rate in stationary state Part II: Multiple Population - Background: cortical populations - spatial coupling - continuum model - stationary state - application: head direction cells

34. Continuum: stationary profile

35. Head direction cells (and line attractor)

36. Neurophysiology of the Rat Hippocampus CA1 CA3 DG electrode Place fields synapses axon soma dendrites pyramidal cells rat brain

37. Hippocampal Place Cells Main property: encoding the animal’s location place field

38. Head-direction Cells r (q) i F q q q i Preferred firing direction Main property: encoding the animal ’s heading

39. Head-direction Cells 90 r (q) i 180 0 300 270 q q i Preferred firing direction Main property: encoding the animal ’s allocentric heading

40. Basic phenomenology Field Equations: Wilson and Cowan, 1972 I: Bump formation A(theta) strong lateral connectivity Possible interpretation of head direction cells: always some cells active  indicate current orientation 0

41. Basic phenomenology Field Equations: Wilson and Cowan, 1972 II. Edge enhancement A(x) Weaker lateral connectivity Possible interpretation of visual cortex cells: contrast enhancement in - orientation - location 0 I(x)

42. Continuum: stationary profile Spiridon&Gerstner Comparison simulation of neurons and macroscopic field equation See: Chapter 9, book: Spiking Neuron Models, W. Gerstner and W. Kistler, 2002

43. Exercises 2,3,4 from 11:15-12:45 Exercise 1: homogeneous stationary solution Exercise 2: bump solution x Exercise 3: bump formation x Blob forms where the input is

44. Field equations and continuum Models for coupled populations Part I: Single Population - Population Activity, definition - high noise - full connectivity - stationary state - mean rate in stationary state Part II: Multiple Population - Background: cortical populations - spatial coupling - continuum model - stationary state Part III: Beware of Pitfalls - oscillations - low noise

45. Book: Spiking Neuron Models. W. Gerstner and W. Kistler Chapter 8 Oscillations

46. Stability of Asynchronous State A(t) fullyconnected coupling J/N linearize h: input potential Search for bifurcation points

47. stable noise Stability of Asynchronous State A(t) 0 for s delay period (Gerstner 2000)

48. noise model A (escape noise/fast noise) high noise I(t) h(t) High-noise activity equation Population activity Membrane potential caused by input Attention: - valid for high noise only, else transients might be wrong - valid for high noise only, else spontaneous oscillations may arise slow transient

49. Random connectivity

50. mean drive variance Random Connectivity/Asynchronous State A(t)=const C inputs per neuron randomlyconnected improved mean field Analogous for column of 1 exc. + 1 inhib. Pop. frequency (single neuron) (Amit&Brunel 1997, Brunel 2000)