1. Lecture 10: Mean Field theory with fluctuations and correlations Reference: A Lerchner et al, Response Variability in Balanced Cortical Networks, q-bio.NC/0402022, Neural Computation 18 634-659 (also on course webpage)

2. Mean field theory for disordered systems In network with fixed randomness (here: random connections):

3. Mean field theory for disordered systems In network with fixed randomness (here: random connections): =>

4. Mean field theory for disordered systems In network with fixed randomness (here: random connections): Different neurons have different number of inputs =>

5. Mean field theory for disordered systems In network with fixed randomness (here: random connections): Different neurons have different number of inputs  Different neurons have different mean input currents =>

6. Mean field theory for disordered systems In network with fixed randomness (here: random connections): Different neurons have different number of inputs  Different neurons have different mean input currents =>

7. Mean field theory for disordered systems In network with fixed randomness (here: random connections): Different neurons have different number of inputs  Different neurons have different mean input currents  Different rates =>

8. Mean field theory for disordered systems In network with fixed randomness (here: random connections): Different neurons have different number of inputs  Different neurons have different mean input currents  Different rates => Temporal fluctuations:

9. Mean field theory for disordered systems In network with fixed randomness (here: random connections): Different neurons have different number of inputs  Different neurons have different mean input currents  Different rates => Have to include these fluctuations in the theory => Temporal fluctuations:

10. Heuristic treatment Start from(one population for now)

11. Heuristic treatment Start from(one population for now)

12. Heuristic treatment Start from Time average: (one population for now)

13. Heuristic treatment Start from Time average: Average over neurons: (one population for now)

14. Heuristic treatment Start from Time average: Average over neurons: Neuron-to-neuron fluctuations: (one population for now)

15. Temporal fluctuations Input current fluctuations:

16. Temporal fluctuations Input current fluctuations: Correlations:

17. Temporal fluctuations Input current fluctuations: Correlations: Have to calculate (“order parameters”) self-consistently

18. 2-population model Like Amit-Brunel model, but with different scaling of synapses (van Vreeswijk-Sompolinsky)

19. 2-population model Like Amit-Brunel model, but with different scaling of synapses (van Vreeswijk-Sompolinsky) Populations 0,1,2 (as before) 0 1 2

20. 2-population model Like Amit-Brunel model, but with different scaling of synapses (van Vreeswijk-Sompolinsky) Populations 0,1,2 (as before) Mean number of connections from population b to a neuron in population a : 0 1 2

21. 2-population model Like Amit-Brunel model, but with different scaling of synapses (van Vreeswijk-Sompolinsky) Populations 0,1,2 (as before) Mean number of connections from population b to a neuron in population a : 0 1 2

22. 2-population model Like Amit-Brunel model, but with different scaling of synapses (van Vreeswijk-Sompolinsky) Populations 0,1,2 (as before) Mean number of connections from population b to a neuron in population a : 0 1 2

23. Means and variances of synaptic strengths

24. Mean:

25. Means and variances of synaptic strengths Mean: Variance:

26. Input current statistics Mean (average of time and neurons)

27. Input current statistics Mean (average of time and neurons)

28. Input current statistics Mean (average of time and neurons) Neuron-to-neuron fluctuations of the temporal mean:

29. Input current statistics Mean (average of time and neurons) Neuron-to-neuron fluctuations of the temporal mean:

30. Input current statistics Mean (average of time and neurons) Neuron-to-neuron fluctuations of the temporal mean: Temporal fluctuations:

31. Equivalent single-neuron problem Single neuron (excitatory or inhibitory) driven by input current

32. Equivalent single-neuron problem Single neuron (excitatory or inhibitory) driven by input current

33. Equivalent single-neuron problem Single neuron (excitatory or inhibitory) driven by input current with

34. Equivalent single-neuron problem Single neuron (excitatory or inhibitory) driven by input current with

35. Can combine the two kinds of fluctuations: Consider the total correlation function

36. Can combine the two kinds of fluctuations: Consider the total correlation function

37. Can combine the two kinds of fluctuations: Consider the total correlation function Average over neurons

38. Can combine the two kinds of fluctuations: Consider the total correlation function Average over neurons Total input current fluctuations:

39. Balance condition Total average current:

40. Balance condition Total average current: i.e., with

41. Balance condition Total average current: i.e., with

42. Balance condition Total average current: i.e., with or, defining

43. Balance condition Total average current: i.e., with or, defining

44. Balance condition Total average current: i.e., with or, defining Solution:

45. Balance condition Total average current: i.e., with or, defining Solution: i.e., can solve for mean rates independent of fluctuations/correlations

46. Correlations/fluctuations Have to do it numerically:

47. Correlations/fluctuations Have to do it numerically: Start with r b from balance eqn

48. Correlations/fluctuations Have to do it numerically: Start with r b from balance eqn, initial estimates

49. Correlations/fluctuations Have to do it numerically: Start with r b from balance eqn, initial estimates Generate Gaussian noisy input current with mean

50. Correlations/fluctuations Have to do it numerically: Start with r b from balance eqn, initial estimates Generate Gaussian noisy input current with mean and correlation function

51. Correlations/fluctuations Have to do it numerically: Start with r b from balance eqn, initial estimates Generate Gaussian noisy input current with mean and correlation function Simulate many trials with different realizations of noise, collect statistics (measure r a, q a, C a (t) )

52. Correlations/fluctuations Have to do it numerically: Start with r b from balance eqn, initial estimates Generate Gaussian noisy input current with mean and correlation function Simulate many trials with different realizations of noise, collect statistics (measure r a, q a, C a (t) ) Use these to generate improved noise samples

53. Correlations/fluctuations Have to do it numerically: Start with r b from balance eqn, initial estimates Generate Gaussian noisy input current with mean and correlation function Simulate many trials with different realizations of noise, collect statistics (measure r a, q a, C a (t) ) Use these to generate improved noise samples Simulate again, repeat until input and output order parameters agree

54. When done, compute firing statistics of single neurons: For a single neuron, with effective input current

55. When done, compute firing statistics of single neurons: For a single neuron, with effective input current Have to hold x ab fixed

56. Experimental background: firing statistics Gershon et al, J Neurophysiol 79, 1135-1144 (1998) x x x x x x

57. Experimental background: firing statistics Gershon et al, J Neurophysiol 79, 1135-1144 (1998) Variance > mean x x x x x x

58. Experimental background: firing statistics Gershon et al, J Neurophysiol 79, 1135-1144 (1998) Variance > mean x x x x x x i,e., Fano factor F > 1

59. Fano factors and correlation functions Spike count:

60. Fano factors and correlation functions Spike count: Mean:

61. Fano factors and correlation functions Spike count: Mean: Variance:

62. Fano factors and correlation functions Spike count: Mean: Variance: => Fano factor

63. Model calculations Synaptic matrix:

64. Model calculations Synaptic matrix: J s = 0.375 J s = 0.75 J s = 1.5 Correlation functions: g = 1

65. Interspike interval distributions Js = 1.5 Js = 0.75 J s = 0.375

66. Spike count variance vs mean

67. What controls F? Membrane potential distributions have width ~ J ab = O(1)

68. What controls F? Low post-spike reset voltage: takes time (~  ) to recover from reset Membrane potential distributions have width ~ J ab = O(1)

69. What controls F? Low post-spike reset voltage: takes time (~  ) to recover from reset => F < 1 Membrane potential distributions have width ~ J ab = O(1)

70. What controls F? Low post-spike reset voltage: takes time (~  ) to recover from reset => F < 1 Reset near threshold: initial spread of membrane potential distribution Membrane potential distributions have width ~ J ab = O(1)

71. What controls F? Low post-spike reset voltage: takes time (~  ) to recover from reset => F < 1 Reset near threshold: initial spread of membrane potential distribution => excess early spikes, F > 1 Membrane potential distributions have width ~ J ab = O(1)