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The standard I&F neuron Assume that membrane is passive (RC circuit) in subthreshold range
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Lecture 8: Integrate-and-Fire Neurons
References:
Dayan and Abbott, sect 5.4Gerstner and Kistler, sects 4.1-4.3, 5.5, 5.6, 6.2.1H Tuckwell, Introduction to Theoretical Neurobiology, v. 2
(Cambridge U Press) Ch 9S Redner, A Guide to First-Passage Processes
(Cambridge U Press) sects 3.2, 4.2
The standard I&F neuronAssume that membrane is passive (RC circuit) in subthreshold range
The standard I&F neuron
VIVVgdtdVC ,)( 0
Assume that membrane is passive (RC circuit) in subthreshold range
The standard I&F neuron
VIVVgdtdVC ,)( 0
Assume that membrane is passive (RC circuit) in subthreshold range
When V reaches threshold, spike and reset at V = Vr
The standard I&F neuron
VIVVgdtdVC ,)( 0
Assume that membrane is passive (RC circuit) in subthreshold range
When V reaches threshold, spike and reset at V = Vr
Constant input)0,/1,(ˆ
000 VgRRCIRIVdtdV
Constant input)0,/1,(ˆ
000 VgRRCIRIVdtdV initial condition: V = 0
Constant input)0,/1,(ˆ
000 VgRRCIRIVdtdV
Equilibrium level: (if RI0
initial condition: V = 0
V=RI0
Constant input)0,/1,(ˆ
000 VgRRCIRIVdtdV
)]/exp(1[ˆ)/exp()ˆ)0((ˆ000 tItIVIV
Equilibrium level: (if RI0
initial condition: V = 0
V=RI0
Solution:
Constant input)0,/1,(ˆ
000 VgRRCIRIVdtdV
)]/exp(1[ˆ)/exp()ˆ)0((ˆ000 tItIVIV
Equilibrium level: (if RI0
initial condition: V = 0
V=RI0
Solution:
if RI0 reset to Vr when V
Constant input)0,/1,(ˆ
000 VgRRCIRIVdtdV
)]/exp(1[ˆ)/exp()ˆ)0((ˆ000 tItIVIV
Equilibrium level: (if RI0
initial condition: V = 0
V=RI0
Solution:
if RI0 reset to Vr when V )/exp(1
0̂
tI
(here Vr = 0)
Constant input)0,/1,(ˆ
000 VgRRCIRIVdtdV
)]/exp(1[ˆ)/exp()ˆ)0((ˆ000 tItIVIV
Equilibrium level: (if RI0
initial condition: V = 0
V=RI0
Solution:
if RI0 reset to Vr when V )/exp(1
0̂
tI
0
0
ˆˆ
logI
It
(here Vr = 0)
Constant input)0,/1,(ˆ
000 VgRRCIRIVdtdV
)]/exp(1[ˆ)/exp()ˆ)0((ˆ000 tItIVIV
Equilibrium level: (if RI0
initial condition: V = 0
V=RI0
Solution:
if RI0 reset to Vr when V )/exp(1
0̂
tI
0
0
ˆˆ
logI
It
(here Vr = 0)
Input-output function
0
0
ˆˆ
log
11
IIt
rRate:
Input-output function
0
0
ˆˆ
log
11
IIt
r
rtt
Rate:
With refractory time:
Input-output function
0
0
ˆˆ
log
11
IIt
r
rtt
Rate:
With refractory time:
0
0
ˆˆ
log
1
II
r
r
Input-output function
0
0
ˆˆ
log
11
IIt
r
rtt
Rate:
With refractory time:
0
0
ˆˆ
log
1
II
r
r
r
rms
General time-dependent input
)(exp1)/exp()0()( tRItttdtVtVt
General time-dependent input
)(exp1)/exp()0()( tRItttdtVtVt
(below threshold)
General time-dependent input
)(exp1)/exp()0()( tRItttdtVtVt
(below threshold)
Synaptic input0)( ss VVggV
dtdVC
Leaky membrane + synaptic current:
Synaptic input0)( ss VVggV
dtdVC
pres rg
Leaky membrane + synaptic current:
Synaptic conductance ~ presynaptic rate
Synaptic input0)( ss VVggV
dtdVC
pres rg
Leaky membrane + synaptic current:
Synaptic conductance ~ presynaptic rate
Reduced effective membrane time constant:
Synaptic input0)( ss VVggV
dtdVC
pres rg
seff gg
C
Leaky membrane + synaptic current:
Synaptic conductance ~ presynaptic rate
Reduced effective membrane time constant:
Synaptic input0)( ss VVggV
dtdVC
pres rg
seff gg
C
ssVgI
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
Spike-response model (1):rewriting the I&F neuron
Separate recovery from reset fromresponse to input current
)()()( 10 tVtVtV
Spike-response model (1):rewriting the I&F neuron
Separate recovery from reset fromresponse to input current
V0 describes recovery:
)()()( 10 tVtVtV
Spike-response model (1):rewriting the I&F neuron
rsp VtVVdt
dV )(0
0
Separate recovery from reset fromresponse to input current
V0 describes recovery:
)()()( 10 tVtVtV
Spike-response model (1):rewriting the I&F neuron
rsp VtVVdt
dV )(0
0
Separate recovery from reset fromresponse to input current
V0 describes recovery:
V1 describes response to input:
)()()( 10 tVtVtV
Spike-response model (1):rewriting the I&F neuron
)(11 tRIV
dtdV
rsp VtVVdt
dV )(0
0
Separate recovery from reset fromresponse to input current
V0 describes recovery:
V1 describes response to input:
)()()( 10 tVtVtV
Spike-response model (1):rewriting the I&F neuron
)(11 tRIV
dtdV
rsp VtVVdt
dV )(0
0
Separate recovery from reset fromresponse to input current
V0 describes recovery:
V1 describes response to input:
independent of spiking
)()()( 10 tVtVtV
Spike-response model (1):rewriting the I&F neuron
)(11 tRIV
dtdV
rsp VtVVdt
dV )(0
0
Separate recovery from reset fromresponse to input current
V0 describes recovery:
V1 describes response to input:
independent of spiking
)()()( 10 tVtVtV
)(exp1)(1 tRItttdtVt
Integrated version:
Spike-response Model (2): extension to general kernels
Spike-response Model (2): extension to general kernels
)(0 spttV (including spike itself)
Spike-response Model (2): extension to general kernels
)(0 spttV (including spike itself)
Get shape of from, e.g. HH solution
Spike-response Model (2): extension to general kernels
)(0 spttV
),(exp1sptttttt
(including spike itself)
Get shape of from, e.g. HH solution
Spike-response Model (2): extension to general kernels
)(0 spttV
),(exp1sptttttt
(including spike itself)
can depend on t-tsp
Get shape of from, e.g. HH solution
Spike-response Model (2): extension to general kernels
)(0 spttV
),(exp1sptttttt
(including spike itself)
can depend on t-tsp
t
prespprespsp tttttItttttdtttV ),()(ˆ),()()(
Get shape of from, e.g. HH solution
with synaptic input:
Spike-response Model (2): extension to general kernels
)(0 spttV
),(exp1sptttttt
(including spike itself)
can depend on t-tsp
t
prespprespsp tttttItttttdtttV ),()(ˆ),()()(
Get shape of from, e.g. HH solution
with synaptic input:
tpre: spike times for presynaptic neuron
Spike-response Model (2): extension to general kernels
)(0 spttV
),(exp1sptttttt
(including spike itself)
can depend on t-tsp
t
prespprespsp tttttItttttdtttV ),()(ˆ),()()(
Get shape of from, e.g. HH solution
with synaptic input:
tpre: spike times for presynaptic neuron
(phenomenlogical: dependence on V is replaced by dependenceon t - tsp)
Approximating HHFirst find the threshold
Approximating HH
Then solve HH equation with V initially at rest and )()( 0 tqtI
First find the threshold
(q0 big enough to cause a spike)
Approximating HH
Then solve HH equation with V initially at rest and )()( 0 tqtI
Identify )(])([)( 0 tVttVt rest where )( 0tV
First find the threshold
(q0 big enough to cause a spike)
Approximating HH
Then solve HH equation with V initially at rest and )()( 0 tqtI
Identify )(])([)( 0 tVttVt rest
Then solve HH equation with V initially at rest and )()()( 10 tttqtI
where )( 0tV
First find the threshold
(q0 big enough to cause a spike)
( very small)
Approximating HH
Then solve HH equation with V initially at rest and )()( 0 tqtI
Identify )(])([)( 0 tVttVt rest
Then solve HH equation with V initially at rest and )()()( 10 tttqtI
where )( 0tV
First find the threshold
Identify )()]()([),( 0111 ttttttVtt
(q0 big enough to cause a spike)
( very small)
Approximating HH
Then solve HH equation with V initially at rest and )()( 0 tqtI
Identify )(])([)( 0 tVttVt rest
Then solve HH equation with V initially at rest and )()()( 10 tttqtI
where )( 0tV
First find the threshold
Identify )()]()([),( 0111 ttttttVtt
(q0 big enough to cause a spike)
( very small)
Comparison with full HH
Solid: HH dashed: SRM
Comparison with full HH
Solid: HH dashed: SRM
Solid: HHDotted: from const currentDashed: optimized for time-dependent current
Rate as function of current:
Noisy input
Noisy input
)()( 0 tIItI
Noisy input
)()( 0 tIItI
White noise:
Noisy input
)()( 0 tIItI
White noise:
0)( tI
Noisy input
)()( 0 tIItI
)()()( 2 tttItI
White noise:
0)( tI
Noisy input
)()( 0 tIItI
)()()( 2 tttItI
White noise:
0)( tI
: “noise power”
Leakless I&F neuron)()( 0 tIItI
dtdVC
Leakless I&F neuron)()( 0 tIItI
dtdVC Langevin equation
Leakless I&F neuron)()( 0 tIItI
dtdVC
I0 case: random walk
)()1( tIdtdVC
Langevin equation
Leakless I&F neuron)()( 0 tIItI
dtdVC
I0 case: random walk
)()1( tIdtdVC
ttItdtV
0)()( =>
Langevin equation
Leakless I&F neuron)()( 0 tIItI
dtdVC
I0 case: random walk
)()1( tIdtdVC
ttItdtV
0)()(
0)()(0
t
tItdtV
=>
averages: mean
Langevin equation
Leakless I&F neuron)()( 0 tIItI
dtdVC
I0 case: random walk
)()1( tIdtdVC
ttItdtV
0)()(
0)()(0
t
tItdtV
t
tttdtdtItItdtdtVttt t
2
2
000 0
2 )()()()(
=>
averages: mean mean square displacement
Langevin equation
Leakless I&F neuron)()( 0 tIItI
dtdVC
I0 case: random walk
)()1( tIdtdVC
ttItdtV
0)()(
0)()(0
t
tItdtV
t
tttdtdtItItdtdtVttt t
2
2
000 0
2 )()()()(
=>
averages: mean mean square displacement
distribution:
Langevin equation
Leakless I&F neuron)()( 0 tIItI
dtdVC
I0 case: random walk
)()1( tIdtdVC
ttItdtV
0)()(
0)()(0
t
tItdtV
t
tttdtdtItItdtdtVttt t
2
2
000 0
2 )()()()(
=>
averages: mean mean square displacement
tV
ttVP 2
2
2 2exp
21)|(
distribution:
Langevin equation
Diffusion Fick’s law:
xPDJ
Diffusion Fick’s law: cf Ohm’s law
xPDJ
xVgI
Diffusion Fick’s law: cf Ohm’s law
xPDJ
xVgI
conservation:xJ
tP
Diffusion Fick’s law: cf Ohm’s law
xPDJ
xVgI
conservation:xJ
tP
2
2
xPD
tP
=>
Diffusion Fick’s law: cf Ohm’s law
xPDJ
xVgI
conservation:xJ
tP
2
2
xPD
tP
=> diffusion equation
Diffusion Fick’s law: cf Ohm’s law
xPDJ
xVgI
conservation:xJ
tP
2
2
xPD
tP
=> diffusion equation
initial condition )()0|( xxP
Diffusion Fick’s law: cf Ohm’s law
xPDJ
xVgI
conservation:xJ
tP
2
2
xPD
tP
=> diffusion equation
initial condition )()0|( xxP
Solution:
Dtx
DttxP
4exp
41)|(
2
Comparison:
From Langevin equation:
tV
ttVP 2
2
2 2exp
21)|(
Comparison:
From Langevin equation:
From diffusion equation (with x -> V)
tV
ttVP 2
2
2 2exp
21)|(
DtV
DttVP
4exp
41)|(
2
Comparison:
From Langevin equation:
From diffusion equation (with x -> V)
tV
ttVP 2
2
2 2exp
21)|(
DtV
DttVP
4exp
41)|(
2
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
Dtx
DtDtx
DttxP
4)2(exp
41
4exp
41)|(
22
Diffusion with threshold: method of images
Absorbing boundary at x = : P() = 0
Add a negative source at x = 2
Dtx
DtDtx
DttxP
4)2(exp
41
4exp
41)|(
22
Probability of having been absorbed by time t:
Dtx
DtdxtxPdxtA
4exp
42)|()(
2
Diffusion with threshold: method of images
Absorbing boundary at x = : P() = 0
Add a negative source at x = 2
Dtx
DtDtx
DttxP
4)2(exp
41
4exp
41)|(
22
Probability of having been absorbed by time t:
Dtx
DtdxtxPdxtA
4exp
42)|()(
2
Change of variables:
2exp
22)(
2
2/
ydytADt
Interspike interval density
(first passage time density)
Interspike interval density
(first passage time density)
DttDydy
dtd
dttdAtP
Dt 4exp
42exp
22)()(
2
2/3
2
2/
Interspike interval density
(first passage time density)
DttDydy
dtd
dttdAtP
Dt 4exp
42exp
22)()(
2
2/3
2
2/
Alternatively, from
xxPDJtP )()(
Interspike interval density
(first passage time density)
DttDydy
dtd
dttdAtP
Dt 4exp
42exp
22)()(
2
2/3
2
2/
Alternatively, from
xxPDJtP )()(
DttDDtx
dxd
Dt
Dtx
DtDtx
DtdxdtP
x
x
4exp
44exp
42
4)2(exp
41
4exp
41)(
2
2/3
2
22
A problem: firing rate = 0
Rate = 1/(mean interspike interval)
A problem: firing rate = 0
Rate = 1/(mean interspike interval)
2/30~)(
ttdtdtttPt
Diffusion + drift
No absorbing boundary:
Dtvtx
DttxP
4)(exp
41)|(
2
Diffusion + drift
No absorbing boundary:
Dtvtx
DttxP
4)(exp
41)|(
2
Need a moving image
Dt
vtxDt
C4
)2(exp41 2
Diffusion + drift
No absorbing boundary:
Dtvtx
DttxP
4)(exp
41)|(
2
Need a moving image
Dt
vtxDt
C4
)2(exp41 2
To make P vanish at x , need
Diffusion + drift
No absorbing boundary:
Dtvtx
DttxP
4)(exp
41)|(
2
Need a moving image
Dt
vtxDt
C4
)2(exp41 2
To make P vanish at x , need
DtvtC
Dtvt
4)(exp
4)(exp
22
Diffusion + drift
No absorbing boundary:
Dtvtx
DttxP
4)(exp
41)|(
2
Need a moving image
Dt
vtxDt
C4
)2(exp41 2
To make P vanish at x , need
DtvtC
Dtvt
4)(exp
4)(exp
22 =>
DvC exp
Diffusion + drift
No absorbing boundary:
Dtvtx
DttxP
4)(exp
41)|(
2
Need a moving image
Dt
vtxDt
C4
)2(exp41 2
To make P vanish at x , need
DtvtC
Dtvt
4)(exp
4)(exp
22 =>
DvC exp
Dt
vtxDv
Dtvtx
DttxP
4)2(expexp
4)(exp
41)|(
22
Solution:
ISI distributionFrom
xxPDJtP )()(
ISI distributionFrom
xxPDJtP )()(
Dtvt
tD
DtvtxC
Dtvtx
dxd
DtDtP
x
4)(exp
4
4)2(exp
4)(exp
4)(
2
2/3
22
ISI distributionFrom
xxPDJtP )()(
Dtvt
tD
DtvtxC
Dtvtx
dxd
DtDtP
x
4)(exp
4
4)2(exp
4)(exp
4)(
2
2/3
22
Now all moments of P(t) are finite.
ISI distributionFrom
xxPDJtP )()(
Dtvt
tD
DtvtxC
Dtvtx
dxd
DtDtP
x
4)(exp
4
4)2(exp
4)(exp
4)(
2
2/3
22
Now all moments of P(t) are finite.
Back to the (noise-driven) leaky I&F neuron
)(0 tIIxdtdx
Back to the (noise-driven) leaky I&F neuron
)(0 tIIxdtdx (V -> x, t in units of , I means RI)
Back to the (noise-driven) leaky I&F neuron
)(0 tIIxdtdx (V -> x, t in units of , I means RI)
0)( tI
Back to the (noise-driven) leaky I&F neuron
)(0 tIIxdtdx (V -> x, t in units of , I means RI)
0)( tI )()()( 2 tttItI
Back to the (noise-driven) leaky I&F neuron
)(0 tIIxdtdx (V -> x, t in units of , I means RI)
0)( tI )()()( 2 tttItI
Brownian motion in a potential 202
1 )()( IxxU
Back to the (noise-driven) leaky I&F neuron
)(0 tIIxdtdx (V -> x, t in units of , I means RI)
0)( tI )()()( 2 tttItI
Brownian motion in a potential 202
1 )()( IxxU
Combining drift and diffusion:
Back to the (noise-driven) leaky I&F neuron
)(0 tIIxdtdx (V -> x, t in units of , I means RI)
0)( tI )()()( 2 tttItI
Brownian motion in a potential 202
1 )()( IxxU
Combining drift and diffusion: Fokker-Planck equation
Back to the (noise-driven) leaky I&F neuron
)(0 tIIxdtdx (V -> x, t in units of , I means RI)
0)( tI )()()( 2 tttItI
Brownian motion in a potential 202
1 )()( IxxU
Combining drift and diffusion: Fokker-Planck equation
Diffusive current:
Back to the (noise-driven) leaky I&F neuron
)(0 tIIxdtdx (V -> x, t in units of , I means RI)
0)( tI )()()( 2 tttItI
Brownian motion in a potential 202
1 )()( IxxU
Combining drift and diffusion: Fokker-Planck equation
Diffusive current:x
txPDtxJdiff )|(),(
Back to the (noise-driven) leaky I&F neuron
)(0 tIIxdtdx (V -> x, t in units of , I means RI)
0)( tI )()()( 2 tttItI
Brownian motion in a potential 202
1 )()( IxxU
Combining drift and diffusion: Fokker-Planck equation
Diffusive current:x
txPDtxJdiff )|(),(
Drift (convective) current:
Back to the (noise-driven) leaky I&F neuron
)(0 tIIxdtdx (V -> x, t in units of , I means RI)
0)( tI )()()( 2 tttItI
Brownian motion in a potential 202
1 )()( IxxU
Combining drift and diffusion: Fokker-Planck equation
Diffusive current:x
txPDtxJdiff )|(),(
Drift (convective) current: )|()()|()(),( 0 txPxItxPxvtxJ drift
Fokker-Planck equationNow use conservation/continuity equation:
Fokker-Planck equationNow use conservation/continuity equation: diffdrift JJ
xtP
Fokker-Planck equationNow use conservation/continuity equation: diffdrift JJ
xtP
xtxPDtxPxI
xttxP )|()|()|(
0
________________________________
Fokker-Planck equationNow use conservation/continuity equation: diffdrift JJ
xtP
xtxPDtxPxI
xttxP )|()|()|(
0
________________________________
First term alone describes a probability cloud with its centerdecaying exponentially toward I0
Fokker-Planck equationNow use conservation/continuity equation: diffdrift JJ
xtP
xtxPDtxPxI
xttxP )|()|()|(
0
________________________________
First term alone describes a probability cloud with its centerdecaying exponentially toward I0
Second term alone describes diffusively spreading probability cloud
Looking for stationary solution0
tP
Looking for stationary solution0
tP 0)()(0
xxPDxPxI
dxd
dxdJi.e.
Looking for stationary solution0
tP 0)()(0
xxPDxPxI
dxd
dxdJ
const)()()( 0
xxPDxPxIxJ
i.e.
=>
Looking for stationary solution0
tP 0)()(0
xxPDxPxI
dxd
dxdJ
const)()()( 0
xxPDxPxIxJ
Boundary conditions: sink at firing threshold x source at x = Vr (= 0 here)
i.e.
=>
Looking for stationary solution0
tP 0)()(0
xxPDxPxI
dxd
dxdJ
const)()()( 0
xxPDxPxIxJ
Boundary conditions: sink at firing threshold x source at x = Vr (= 0 here)
xxP ,0)(
i.e.
=>
Looking for stationary solution0
tP 0)()(0
xxPDxPxI
dxd
dxdJ
const)()()( 0
xxPDxPxIxJ
Boundary conditions: sink at firing threshold x source at x = Vr (= 0 here)
xxP ,0)( 0and,0)( xxxJ
i.e.
=>
Looking for stationary solution0
tP 0)()(0
xxPDxPxI
dxd
dxdJ
const)()()( 0
xxPDxPxIxJ
Boundary conditions: sink at firing threshold x source at x = Vr (= 0 here)
xxP ,0)( 0and,0)( xxxJ
i.e.
=>
Firing rate: current out at threshold:
Looking for stationary solution0
tP 0)()(0
xxPDxPxI
dxd
dxdJ
const)()()( 0
xxPDxPxIxJ
Boundary conditions: sink at firing threshold x source at x = Vr (= 0 here)
xxP ,0)( 0and,0)( xxxJ
xdxdPDJr )(
i.e.
=>
Firing rate: current out at threshold:
Looking for stationary solution0
tP 0)()(0
xxPDxPxI
dxd
dxdJ
const)()()( 0
xxPDxPxIxJ
Boundary conditions: sink at firing threshold x source at x = Vr (= 0 here)
xxP ,0)( 0and,0)( xxxJ
xdxdPDJr )(
i.e.
=>
Firing rate: current out at threshold: = reinjection rate at reset:
Looking for stationary solution0
tP 0)()(0
xxPDxPxI
dxd
dxdJ
const)()()( 0
xxPDxPxIxJ
Boundary conditions: sink at firing threshold x source at x = Vr (= 0 here)
xxP ,0)( 0and,0)( xxxJ
xdxdPDJr )(
00 )0()0(
xdxdPDPIJr
i.e.
=>
Firing rate: current out at threshold: = reinjection rate at reset:
Stationary solution (2)
Also need normalization:
Stationary solution (2)
Also need normalization:
1)(xPdx
Stationary solution (2)
Also need normalization:
1)(xPdx
Below reset level, J :
Stationary solution (2)
Also need normalization:
1)(xPdx
Below reset level, J : 0)()(0
xxPDxPxI
Stationary solution (2)
Also need normalization:
1)(xPdx
Below reset level, J : 0)()(0
xxPDxPxI
has solution
DIxcxP
2)(exp)(
20
1
Stationary solution (2)
Also need normalization:
1)(xPdx
Below reset level, J : 0)()(0
xxPDxPxI
has solution
DIxcxP
2)(exp)(
20
1
x DIydy
DIxcxP
2)(exp
2)(exp)(
20
20
2Between rest and threshold:
Stationary solution (2)
Also need normalization:
1)(xPdx
Below reset level, J : 0)()(0
xxPDxPxI
has solution
DIxcxP
2)(exp)(
20
1
x DIydy
DIxcxP
2)(exp
2)(exp)(
20
20
2Between rest and threshold:
B.C. at x :
Stationary solution (2)
Also need normalization:
1)(xPdx
Below reset level, J : 0)()(0
xxPDxPxI
has solution
DIxcxP
2)(exp)(
20
1
x DIydy
DIxcxP
2)(exp
2)(exp)(
20
20
2Between rest and threshold:
B.C. at x : 2
20
20
2 2)(exp
2)(exp Dc
DI
DIDc
dxdPDr
x
Stationary solution (2)
Also need normalization:
1)(xPdx
Below reset level, J : 0)()(0
xxPDxPxI
has solution
DIxcxP
2)(exp)(
20
1
x DIydy
DIxcxP
2)(exp
2)(exp)(
20
20
2Between rest and threshold:
B.C. at x : 2
20
20
2 2)(exp
2)(exp Dc
DI
DIDc
dxdPDr
x
=>Drc 2
Stationary solution (3)Continuity at x = =>
Stationary solution (3)Continuity at x = =>
0
20
20
2
20
1 2)(exp
2exp
2exp
DIydy
DIc
DIc
Stationary solution (3)Continuity at x = =>
0
20
20
2
20
1 2)(exp
2exp
2exp
DIydy
DIc
DIc
i.e.,
0
20
21 2)(exp
DIydycc
Stationary solution (3)Continuity at x = =>
0
20
20
2
20
1 2)(exp
2exp
2exp
DIydy
DIc
DIc
i.e.,
0
20
21 2)(exp
DIydycc
algebra … =>
Stationary solution (3)Continuity at x = =>
0
20
20
2
20
1 2)(exp
2exp
2exp
DIydy
DIc
DIc
i.e.,
0
20
21 2)(exp
DIydycc
algebra … =>
)erf1)(exp(
1
)erf1)(exp(
12/)(
/22/)(
2/
0
0
0
0
xxdxxxdxr I
I
DI
DI
Stationary solution (3)Continuity at x = =>
0
20
20
2
20
1 2)(exp
2exp
2exp
DIydy
DIc
DIc
i.e.,
0
20
21 2)(exp
DIydycc
algebra … =>
)erf1)(exp(
1
)erf1)(exp(
12/)(
/22/)(
2/
0
0
0
0
xxdxxxdxr I
I
DI
DI
with refractory timer
Stationary solution (3)Continuity at x = =>
0
20
20
2
20
1 2)(exp
2exp
2exp
DIydy
DIc
DIc
i.e.,
0
20
21 2)(exp
DIydycc
algebra … =>
)erf1)(exp(
1
)erf1)(exp(
12/)(
/22/)(
2/
0
0
0
0
xxdxxxdxr I
I
DI
DI
with refractory timer)erf1)(exp(
12/)(
/
0
0
xxdxr I
Ir
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 V Rate vs mean input current
(for several noise power levels)
