1. Lecture 8: Integrate-and-Fire Neurons References: Dayan and Abbott, sect 5.4 Gerstner and Kistler, sects 4.1-4.3, 5.5, 5.6, 6.2.1 H Tuckwell, Introduction to Theoretical Neurobiology, v. 2 (Cambridge U Press) Ch 9 S Redner, A Guide to First-Passage Processes (Cambridge U Press) sects 3.2, 4.2

2. The standard I&F neuron Assume that membrane is passive (RC circuit) in subthreshold range

3. The standard I&F neuron Assume that membrane is passive (RC circuit) in subthreshold range

4. The standard I&F neuron Assume that membrane is passive (RC circuit) in subthreshold range When V reaches threshold, spike and reset at V = V r

5. The standard I&F neuron Assume that membrane is passive (RC circuit) in subthreshold range When V reaches threshold, spike and reset at V = V r

6. Constant input

7. initial condition: V = 0

8. Constant input Equilibrium level: (if RI 0  initial condition: V = 0 V=RI 0

9. Constant input Equilibrium level: (if RI 0  initial condition: V = 0 V=RI 0 Solution:

10. Constant input Equilibrium level: (if RI 0  initial condition: V = 0 V=RI 0 Solution: if RI 0  reset to V r when V 

11. Constant input Equilibrium level: (if RI 0  initial condition: V = 0 V=RI 0 Solution: if RI 0  reset to V r when V  (here V r = 0 )

12. Constant input Equilibrium level: (if RI 0  initial condition: V = 0 V=RI 0 Solution: if RI 0  reset to V r when V   (here V r = 0 )

13. Constant input Equilibrium level: (if RI 0  initial condition: V = 0 V=RI 0 Solution: if RI 0  reset to V r when V   (here V r = 0 )

14. Input-output function Rate:

15. Input-output function Rate: With refractory time:

16. Input-output function Rate: With refractory time:

17. Input-output function Rate: With refractory time:  r   r  ms

18. General time-dependent input

19. (below threshold)

20. General time-dependent input (below threshold)

21. Synaptic input Leaky membrane + synaptic current:

22. Synaptic input Leaky membrane + synaptic current: Synaptic conductance ~ presynaptic rate

23. Synaptic input Leaky membrane + synaptic current: Synaptic conductance ~ presynaptic rate Reduced effective membrane time constant:

24. Synaptic input Leaky membrane + synaptic current: Synaptic conductance ~ presynaptic rate Reduced effective membrane time constant:

25. Synaptic input Leaky membrane + synaptic current: Synaptic conductance ~ presynaptic rate Reduced effective membrane time constant: and effective current input

26. Spike-response model (1): rewriting the I&F neuron

27. Separate recovery from reset from response to input current

28. Spike-response model (1): rewriting the I&F neuron Separate recovery from reset from response to input current V 0 describes recovery:

29. Spike-response model (1): rewriting the I&F neuron Separate recovery from reset from response to input current V 0 describes recovery:

30. Spike-response model (1): rewriting the I&F neuron Separate recovery from reset from response to input current V 0 describes recovery: V 1 describes response to input:

31. Spike-response model (1): rewriting the I&F neuron Separate recovery from reset from response to input current V 0 describes recovery: V 1 describes response to input:

32. Spike-response model (1): rewriting the I&F neuron Separate recovery from reset from response to input current V 0 describes recovery: V 1 describes response to input: independent of spiking

33. Spike-response model (1): rewriting the I&F neuron Separate recovery from reset from response to input current V 0 describes recovery: V 1 describes response to input: independent of spiking Integrated version:

34. Spike-response Model (2): extension to general kernels

35. (including spike itself)

36. Spike-response Model (2): extension to general kernels (including spike itself) Get shape of  from, e.g. HH solution

37. Spike-response Model (2): extension to general kernels (including spike itself) Get shape of  from, e.g. HH solution

38. Spike-response Model (2): extension to general kernels (including spike itself) can depend on t-t sp Get shape of  from, e.g. HH solution

39. Spike-response Model (2): extension to general kernels (including spike itself) can depend on t-t sp Get shape of  from, e.g. HH solution with synaptic input:

40. Spike-response Model (2): extension to general kernels (including spike itself) can depend on t-t sp Get shape of  from, e.g. HH solution with synaptic input: t pre : spike times for presynaptic neuron

41. Spike-response Model (2): extension to general kernels (including spike itself) can depend on t-t sp Get shape of  from, e.g. HH solution with synaptic input: t pre : spike times for presynaptic neuron (phenomenlogical: dependence on V is replaced by dependence on t - t sp )

42. Approximating HH First find the threshold 

43. Approximating HH Then solve HH equation with V initially at rest and First find the threshold  ( q 0 big enough to cause a spike)

44. Approximating HH Then solve HH equation with V initially at rest and Identifywhere First find the threshold  ( q 0 big enough to cause a spike)

45. Approximating HH Then solve HH equation with V initially at rest and Identify Then solve HH equation with V initially at rest and where First find the threshold  ( q 0 big enough to cause a spike) (  very small)

46. Approximating HH Then solve HH equation with V initially at rest and Identify Then solve HH equation with V initially at rest and where First find the threshold  Identify ( q 0 big enough to cause a spike) (  very small)

47. Approximating HH Then solve HH equation with V initially at rest and Identify Then solve HH equation with V initially at rest and where First find the threshold  Identify ( q 0 big enough to cause a spike) (  very small)

48. Comparison with full HH Solid: HHdashed: SRM

49. Comparison with full HH Solid: HHdashed: SRM Solid: HH Dotted:  from const current Dashed:  optimized for time-dependent current Rate as function of current:

50. Noisy input

52. White noise:

53. Noisy input White noise:

54. Noisy input White noise:

55. Noisy input White noise:   : “noise power”

56. Leakless I&F neuron

57. Langevin equation

58. Leakless I&F neuron I 0  case: random walk Langevin equation

59. Leakless I&F neuron I 0  case: random walk => Langevin equation

60. Leakless I&F neuron I 0  case: random walk => averages: mean Langevin equation

61. Leakless I&F neuron I 0  case: random walk => averages: meanmean square displacement Langevin equation

62. Leakless I&F neuron I 0  case: random walk => averages: meanmean square displacement distribution: Langevin equation

63. Leakless I&F neuron I 0  case: random walk => averages: meanmean square displacement distribution: Langevin equation

64. Diffusion Fick’s law:

65. Diffusion Fick’s law:cf Ohm’s law

66. Diffusion Fick’s law:cf Ohm’s law conservation:

67. Diffusion Fick’s law:cf Ohm’s law conservation: =>

68. Diffusion Fick’s law:cf Ohm’s law conservation: => diffusion equation

69. Diffusion Fick’s law:cf Ohm’s law conservation: => diffusion equation initial condition

70. Diffusion Fick’s law:cf Ohm’s law conservation: => diffusion equation initial condition Solution:

71. Comparison: From Langevin equation:

72. Comparison: From Langevin equation: From diffusion equation (with x -> V )

73. Comparison: From Langevin equation: From diffusion equation (with x -> V )  identify  2 = 2D

74. Diffusion with threshold: method of images Absorbing boundary at x =  : P(  ) = 0

75. Diffusion with threshold: method of images Absorbing boundary at x =  : P(  ) = 0 Add a negative source at x = 2 

76. Diffusion with threshold: method of images Absorbing boundary at x =  : P(  ) = 0 Add a negative source at x = 2 

77. Diffusion with threshold: method of images Absorbing boundary at x =  : P(  ) = 0 Add a negative source at x = 2  Probability of having been absorbed by time t :

78. Diffusion with threshold: method of images Absorbing boundary at x =  : P(  ) = 0 Add a negative source at x = 2  Probability of having been absorbed by time t : Change of variables:

79. Interspike interval density (first passage time density)

80. Interspike interval density (first passage time density)

81. Interspike interval density (first passage time density) Alternatively, from

82. Interspike interval density (first passage time density) Alternatively, from

83. A problem: firing rate = 0 Rate = 1/(mean interspike interval)

84. A problem: firing rate = 0 Rate = 1/(mean interspike interval)

85. Diffusion + drift No absorbing boundary:

86. Diffusion + drift No absorbing boundary: Need a moving image

87. Diffusion + drift No absorbing boundary: Need a moving image To make P vanish at x , need

88. Diffusion + drift No absorbing boundary: Need a moving image To make P vanish at x , need

89. Diffusion + drift No absorbing boundary: Need a moving image To make P vanish at x , need =>

90. Diffusion + drift No absorbing boundary: Need a moving image To make P vanish at x , need => Solution:

91. ISI distribution From

92. ISI distribution From

93. ISI distribution From Now all moments of P(t) are finite.

94. ISI distribution From Now all moments of P(t) are finite.

95. Back to the (noise-driven) leaky I&F neuron

96. ( V -> x, t in units of , I means RI )

97. Back to the (noise-driven) leaky I&F neuron ( V -> x, t in units of , I means RI )

98. Back to the (noise-driven) leaky I&F neuron ( V -> x, t in units of , I means RI )

99. Back to the (noise-driven) leaky I&F neuron ( V -> x, t in units of , I means RI ) Brownian motion in a potential

100. Back to the (noise-driven) leaky I&F neuron ( V -> x, t in units of , I means RI ) Brownian motion in a potential Combining drift and diffusion:

101. Back to the (noise-driven) leaky I&F neuron ( V -> x, t in units of , I means RI ) Brownian motion in a potential Combining drift and diffusion: Fokker-Planck equation

102. Back to the (noise-driven) leaky I&F neuron ( V -> x, t in units of , I means RI ) Brownian motion in a potential Combining drift and diffusion: Fokker-Planck equation Diffusive current:

103. Back to the (noise-driven) leaky I&F neuron ( V -> x, t in units of , I means RI ) Brownian motion in a potential Combining drift and diffusion: Fokker-Planck equation Diffusive current:

104. Back to the (noise-driven) leaky I&F neuron ( V -> x, t in units of , I means RI ) Brownian motion in a potential Combining drift and diffusion: Fokker-Planck equation Diffusive current: Drift (convective) current:

105. Back to the (noise-driven) leaky I&F neuron ( V -> x, t in units of , I means RI ) Brownian motion in a potential Combining drift and diffusion: Fokker-Planck equation Diffusive current: Drift (convective) current:

106. Fokker-Planck equation Now use conservation/continuity equation:

107. Fokker-Planck equation Now use conservation/continuity equation:

108. Fokker-Planck equation Now use conservation/continuity equation: ________________________________

109. Fokker-Planck equation Now use conservation/continuity equation: ________________________________ First term alone describes a probability cloud with its center decaying exponentially toward I 0

110. Fokker-Planck equation Now use conservation/continuity equation: ________________________________ First term alone describes a probability cloud with its center decaying exponentially toward I 0 Second term alone describes diffusively spreading probability cloud

111. Looking for stationary solution

112. i.e.

113. Looking for stationary solution i.e. =>

114. Looking for stationary solution Boundary conditions: sink at firing threshold x  source at x = V r ( = 0 here) i.e. =>

115. Looking for stationary solution Boundary conditions: sink at firing threshold x  source at x = V r ( = 0 here) i.e. =>

116. Looking for stationary solution Boundary conditions: sink at firing threshold x  source at x = V r ( = 0 here) i.e. =>

117. Looking for stationary solution Boundary conditions: sink at firing threshold x  source at x = V r ( = 0 here) i.e. => Firing rate: current out at threshold:

118. Looking for stationary solution Boundary conditions: sink at firing threshold x  source at x = V r ( = 0 here) i.e. => Firing rate: current out at threshold:

119. Looking for stationary solution Boundary conditions: sink at firing threshold x  source at x = V r ( = 0 here) i.e. => Firing rate: current out at threshold:= reinjection rate at reset:

120. Looking for stationary solution Boundary conditions: sink at firing threshold x  source at x = V r ( = 0 here) i.e. => Firing rate: current out at threshold:= reinjection rate at reset:

121. Stationary solution (2) Also need normalization:

122. Stationary solution (2) Also need normalization:

123. Stationary solution (2) Also need normalization: Below reset level, J  :

124. Stationary solution (2) Also need normalization: Below reset level, J  :

125. Stationary solution (2) Also need normalization: Below reset level, J  : has solution

126. Stationary solution (2) Also need normalization: Below reset level, J  : has solution Between rest and threshold:

127. Stationary solution (2) Also need normalization: Below reset level, J  : has solution Between rest and threshold: B.C. at x  :

128. Stationary solution (2) Also need normalization: Below reset level, J  : has solution Between rest and threshold: B.C. at x  :

129. Stationary solution (2) Also need normalization: Below reset level, J  : has solution Between rest and threshold: B.C. at x  : =>

130. Stationary solution (3) Continuity at x =  =>

131. Stationary solution (3) Continuity at x =  =>

132. Stationary solution (3) Continuity at x =  => i.e.,

133. Stationary solution (3) Continuity at x =  => i.e., algebra … =>

134. Stationary solution (3) Continuity at x =  => i.e., algebra … =>

135. Stationary solution (3) Continuity at x =  => i.e., algebra … => with refractory time  r

136. Stationary solution (3) Continuity at x =  => i.e., algebra … => with refractory time  r

137. membrane potential histories, distributions; rate vs input Histories of V

138. membrane potential histories, distributions; rate vs input Histories of V Distributions of V (for several noise power levels)

139. membrane potential histories, distributions; rate vs input Histories of V Distributions of VRate vs mean input current (for several noise power levels)