Lecture 8: Integrate-and-Fire Neurons References: Dayan and Abbott, sect 5.4 Gerstner and Kistler, sects , 5.5, 5.6, 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

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

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

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

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

Constant input

initial condition: V = 0

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

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

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

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 )

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 )

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 )

Input-output function Rate:

Input-output function Rate: With refractory time:

Input-output function Rate: With refractory time:

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

General time-dependent input

(below threshold)

General time-dependent input (below threshold)

Synaptic input Leaky membrane + synaptic current:

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

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

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

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

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

Separate recovery from reset from response to input current

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

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

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:

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:

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

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:

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

(including spike itself)

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

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

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

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:

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

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 )

Approximating HH First find the threshold 

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

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)

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)

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)

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)

Comparison with full HH Solid: HHdashed: SRM

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

Noisy input

White noise:

Noisy input White noise:

Noisy input White noise:

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

Leakless I&F neuron

Langevin equation

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

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

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

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

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

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

Diffusion Fick’s law:

Diffusion Fick’s law:cf Ohm’s law

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

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

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

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

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

Comparison: From Langevin equation:

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

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

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

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

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

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 :

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:

Interspike interval density (first passage time density)

Interspike interval density (first passage time density)

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

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

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

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

Diffusion + drift No absorbing boundary:

Diffusion + drift No absorbing boundary: Need a moving image

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

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

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

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

ISI distribution From

ISI distribution From

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

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

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

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

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

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

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

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:

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

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:

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:

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:

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:

Fokker-Planck equation Now use conservation/continuity equation:

Fokker-Planck equation Now use conservation/continuity equation:

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

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

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

Looking for stationary solution

i.e.

Looking for stationary solution i.e. =>

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

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

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

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:

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:

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:

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:

Stationary solution (2) Also need normalization:

Stationary solution (2) Also need normalization:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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