1. Mean Field Theories in Neuroscience B. Cessac, Neuromathcomp, INRIA

5. Cortical columns.

7. Small cylinders, of diameter 0.1~1mm, crossing cortex layers, with about 10 3 -10 4 neurons, from different types, strongly connected.

8. Cortical columns are involved in elementary sensori-motor like vision. Cortical columns.

9. They are composed of neurons belonging to a small number of populations interacting together. These populations belong to different cortex layers. Deep layers Layer IV Superficial layers Pyramidal Cells Inhibitory Cells Spiny Stellate Cells Thalamic Input Cortical columns.

10. Deep layers Layer IV Superficial layers Pyramidal Cells Inhibitory Cells Spiny Stellate Cells Thalamic Input It is possible and useful to propose phenomenological models characterizing the mesoscopic activity of cortical columns, predicting the behaviour of local field potential generated by the electric activity of neurons and to relate this behavior with measures and clinical observations (epilepsy). Cortical columns.

11. Cortical column paradigm Type of cortical column Definition Spatial scale Number of neurons Micro-column Mini-column Orientation column Macro-column or Hyper-column (V1) Neural Mass 40-50 µm200-300 µm 600 µm (and more) 10 mm 80-100 neuronsSeveral mini-columns 60-100 mini-columns 10000 neurons 100XThousand neurons of the same type (pyr, stellate,…) OI pixel Our Column 50-100 µm 150-200 neurons Functional Anatomical Physico-functional Cortical Area Courtesy. S. Chemla.

12. Mean field models.

13. Neurons and synapses.

16. Pre-synaptic neuron j Post-synaptic neuron i

17. Neurons and synapses. Pre-synaptic neuron j Post-synaptic neuron i

18. Neurons and synapses. Pre-synaptic neuron j Post-synaptic neuron i

19. Neurons and synapses. Pre-synaptic neuron j Post-synaptic neuron i

20. Neurons and synapses. Pre-synaptic neuron j Post-synaptic neuron i

21. Neurons and synapses. Pre-synaptic neuron j Post-synaptic neuron i

22. Neurons and synapses. Pre-synaptic neuron j Post-synaptic neuron i

23. Neurons and synapses. Pre-synaptic neuron j Post-synaptic neuron i

24. Neurons and synapses. Pre-synaptic neuron j Post-synaptic neuron i

25. Neural mass model.

26. P populations of neurons, a =1... P

27. Neural mass model. Voltage-based model P populations of neurons, a =1... P

28. Neural mass model. Voltage-based model P populations of neurons, a =1... P Assumptions:

29. Neural mass model. Voltage-based model P populations of neurons, a =1... P Assumptions: Synapse response, current and noise depend only on the neuronal population.

30. Neural mass model. Voltage-based model P populations of neurons, a =1... P Assumptions: Synapse response, current and noise depend only on the neuronal population. 1 2 3

31. Neural mass model. Voltage-based model P populations of neurons, a =1... P W 12 1 2 3 W 32 W 23 W 31 W 21 Assumptions: Synapse response, current and noise depend only on the neuronal population.

32. Neural mass model. Voltage-based model P populations of neurons, a =1... P Synaptic weights Assumptions: Synapse response, current and noise depend only on the neuronal population. The probability distribution of synaptic efficacies depend only on pre- and post synaptic neuron' population W 12 1 2 3 W 32 W 23 W 31 W 21 (independent)‏

33. Neural mass model. Voltage-based model P populations of neurons, a =1... P Assumptions: Synapse response, current and noise depend only on the neuronal population. The probability distribution of synaptic efficacies depend only on pre- and post synaptic neuron' population Synaptic weights (independent)‏

35. Dynamic mean-field theory.

36. Voltage-based model

37. Dynamic mean-field theory. Voltage-based model Stochastic (annealed)‏ Random (quenched)‏ Nonlinear

38. Dynamic mean-field theory. Voltage-based model

39. Dynamic mean-field theory. Voltage-based model

40. Dynamic mean-field theory. Voltage-based model Local interactions field.

41. Dynamic mean-field theory. Voltage-based model Local interactions field.

42. Dynamic mean-field theory. Voltage-based model Local interactions field. Non random synaptic weights.

43. Dynamic mean-field theory. Voltage-based model Local interactions field. Non random synaptic weights.

44. Dynamic mean-field theory. Voltage-based model Local interactions field. Non random synaptic weights.

45. Dynamic mean-field theory. Voltage-based model Local interactions field. Non random synaptic weights.

46. Dynamic mean-field theory. Voltage-based model Local interactions field. Non random synaptic weights. Dynamic mean-field equations.

47. Dynamic mean-field theory. Voltage-based model Local interactions field. Non random synaptic weights. Dynamic mean-field equations.

48. Dynamic mean-field theory. Voltage-based model Non random synaptic weights. Naive mean-field equations. Local interactions field.

49. Dynamic mean-field theory. Voltage-based model Non random synaptic weights. Naive mean-field equations. Local interactions field.

50. Dynamic mean-field theory. Voltage-based model Non random synaptic weights. Naive mean-field equations. Local interactions field.

51. Dynamic mean-field theory. Voltage-based model Non random synaptic weights. Naive mean-field equations. Local interactions field. ?

52. Jansen-Ritt equations.

54. It is possible and useful to propose phenomenological models characterizing the mesoscopique activity of cortical columns, predicting the behaviour of local field potential generated by the electric activity of neurons and to relate this behavior with measures and clinical observations (epilepsy).

55. Jansen-Ritt equations. It is possible and useful to propose phenomenological models characterizing the mesoscopique activity of cortical columns, predicting the behaviour of local field potential generated by the electric activity of neurons and to relate this behavior with measures and clinical observations (epilepsy).

56. Jansen – Rit model (1995). Jansen-Ritt equations. It is possible and useful to propose phenomenological models characterizing the mesoscopique activity of cortical columns, predicting the behaviour of local field potential generated by the electric activity of neurons and to relate this behavior with measures and clinical observations (epilepsy).

57. Jansen – Rit model (1995). Jansen-Ritt equations. It is possible and useful to propose phenomenological models characterizing the mesoscopique activity of cortical columns, predicting the behaviour of local field potential generated by the electric activity of neurons and to relate this behavior with measures and clinical observations (epilepsy).

58. Bifurcation analysis Eigenvalues of the Jacobian of the dynamic system when the thalamic entry varies Some bifurcations of the system The bifurcations can predict neural behaviours Grimbert, Faugeras. Neural Computation 2006

62. Why is it working so well ?

63. Dynamic mean-field theory. Voltage-based model Local interaction field.

64. Dynamic mean-field theory. Voltage-based model Local interaction field. Random synaptic weights.

65. Dynamic mean-field theory. Voltage-based model Local interaction field. Random synaptic weights. Prop 1.  i is V-conditionally Gaussian with mean and covariance.

66. Dynamic mean-field theory. Voltage-based model Local interaction field. Random synaptic weights. DMFT

67. Dynamic mean-field theory. Voltage-based model Local interaction field. Random synaptic weights. DMFT where U ab is a Gaussian process with mean and covariance

68. Dynamic mean-field theory. Voltage-based model Local interaction field. Random synaptic weights. DMFT where U ab is a Gaussian process with mean and covariance Dynamic mean-field equations.

69. Dynamic mean-field theory. Simple model Random synaptic weights.

70. Dynamic mean-field theory. Simple model Dynamic mean-field equations. Random synaptic weights.

71. Dynamic mean-field theory. Simple model Dynamic mean-field equations. Random synaptic weights.

72. Dynamic mean-field theory. Simple model Dynamic mean-field equations. Random synaptic weights.

73. Dynamic mean-field theory. Simple model Dynamic mean-field equations. Random synaptic weights.

74. Dynamic mean-field theory. Simple model Dynamic mean-field equations. Random synaptic weights.

75. Dynamic mean-field theory. Simple model Dynamic mean-field equations. Random synaptic weights.

76. Dynamic mean-field theory. Simple model Dynamic mean-field equations. Random synaptic weights.

77. Dynamic mean-field theory. Simple model Dynamic mean-field equations. Random synaptic weights.

78. Dynamic mean-field theory. Simple model Non random synaptic weights. Dynamic mean-field equations.

79. Dynamic mean-field theory. Simple model Non random synaptic weights. Dynamic mean-field equations.

80. Dynamic mean-field theory. Simple model Non random synaptic weights. Dynamic mean-field equations.

81. Dynamic mean-field theory. Simple model Non random synaptic weights. Dynamic mean-field equations.

82. Dynamic mean-field theory. Simple model Non random synaptic weights. Dynamic mean-field equations.

83. Dynamic mean-field theory. Simple model Non random synaptic weights. Dynamic mean-field equations.

84. Dynamic mean-field theory. Simple model Non random synaptic weights. Dynamic mean-field equations.

85. Dynamic mean-field theory. Simple model Non random synaptic weights. Dynamic mean-field equations. Naive mean-field equations.

86. Dynamic mean-field theory. Simple model Dynamic mean-field equations. Random synaptic weights.

87. Dynamic mean-field theory. Simple model Dynamic mean-field equations. Random synaptic weights. Non Markovian process, depending on the whole past.

88. Dynamic mean-field theory. Simple model Dynamic mean-field equations. Random synaptic weights. Non Markovian process, depending on the whole past. Evolution is not ruled anymore by EDOs but by a mapping on a space of trajectories.

89. Dynamic mean-field theory. Voltage-based model Local interaction field. Random synaptic weights. Dynamic mean-field equations. Th. (Faugeras, Touboul, Cessac, 2008)‏

90. Dynamic mean-field theory. Voltage-based model Local interaction field. Random synaptic weights. Dynamic mean-field equations. Th. (Faugeras, Touboul, Cessac, 2008)‏ Existence and uniqueness of solutions in finite time. Existence and uniqueness of solutions in finite time.

91. Dynamic mean-field theory. Voltage-based model Local interaction field. Random synaptic weights. Dynamic mean-field equations. Th. (Faugeras, Touboul, Cessac, 2008)‏ Existence and uniqueness of solutions in finite time. Existence and uniqueness of solutions in finite time. Existence and uniqueness of stationary solutions in a specific region of the macroscopic parameters space. Existence and uniqueness of stationary solutions in a specific region of the macroscopic parameters space.

92. Dynamic mean-field theory. Voltage-based model Local interaction field. Random synaptic weights. Dynamic mean-field equations. Th. (Faugeras, Touboul, Cessac, 2008)‏ Existence and uniqueness of solutions in finite time. Existence and uniqueness of solutions in finite time. Existence and uniqueness of stationary solutions in a specific region of the macroscopic parameters space. Existence and uniqueness of stationary solutions in a specific region of the macroscopic parameters space. Constructive proof => Simulation algorithm. Constructive proof => Simulation algorithm.

94. Are there new and measurable effects here ?

95. How to study DMFT equations and their bifurcations ?

96. Are there new and measurable effects here ? How to study DMFT equations and their bifurcations ? What is synaptic weights are correlated ?