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

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

4. Population - 50 000 neurons - 20 percent inhibitory - randomly connected Signal transmission in populations of neurons 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

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

6. High-noise activity equation 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,

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

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

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

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

11. Exercises 1, now Next lecture 10:15 Exercise 1: homogeneous stationary solution fully connected N >> 1 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 fully connected coupling J/N input potential frequency (single neuron) blackboard Homogeneous network All neurons are identical, Single neuron rate = population rate A(t)= A 0 = const Single neuron

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

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

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

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

18. High-noise activity equation 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

19. 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 Wulfram Gerstner EPFL

20. Microscopic vs. Macroscopic Coupling I(t)

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

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

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 Orientation selective 0 rate 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 Orientation selective 0 rate Stimulus orientation Cell 1

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

31. Continuous Networks

32. Several populations Continuum Blackboard

33. 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 Field equations and continuum Models for coupled populations

34. Continuum: stationary profile

35. Head direction cells (and line attractor)

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

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

38. Head-direction Cells Main property: encoding the animal ’s heading   Preferred firing direction r (  i   i

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

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

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

42. 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

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

44. 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 Field equations and continuum Models for coupled populations

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

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

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

48. High-noise activity equation noise model A I(t) h(t) (escape noise/fast noise) high noise slow transient 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

49. Random connectivity

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

51. The end

52. Liquid state/ Information buffering

53. Ultra-short-term information buffering (‘echo’) I(t) How much information in out(t) about signal ? (Maass et al., 2002, H.Jäger 2004) 640 excitatory n. 160 inhibitory n. 20ms time constant 50 connections/n. population activity bias N parameters

54. Ultra-short-term information buffering (‘echo’) I(t) (Mayor&Gerstner) bias mean-field readout microscopic readout