Networks of Integrate–and–Fire Cells
We follow Dayan & Abbott, Theoretical Neuroscience, §5.5, and, starting from V (0) =
EL , solve
τm V 0 (t) = EL − V (t) + Rm I(t)
until V (t) reaches Vth , at which time we reset V to rest and resume. We have coded this
in iaf1.m and included a representative run below.
iaf1(6,−50,0.05,10)
10
0
−10
V (mV)
−20
−30
−40
−50
−60
−70
0
1
2
3
4
5
6
7
8
9
10
t (ms)
Figure 1. See iaf1.m for parameter values.
It is customary to include an additional dynamical conductance with a potassium
reversal potential that suffices to mimic spike rate adaptation. Starting from V (0) = EL
and gsra (0) = 0 we solve
τm V 0 (t) = EL − V (t) + Rm I(t) − rm gsra (t)(V (t) − EK )
0
τsra gsra
(t) = −gsra (t)
until V (t) reaches Vth , at which time we reset V and increment gsra and resume. We have
coded this in iaf2.m and included a representative run below.
1
iaf2(10,−50,0.025,30)
10
7
0
6
−10
gsra (mS/cm )
5
V (mV)
2
−20
−30
−40
4
3
2
−50
1
−60
−70
0
10
20
0
30
0
10
t (ms)
20
30
t (ms)
Figure 2. See iaf2.m for parameter values.
We now suppose we have a network of cells and denote the potentials by V1 through
VN and their associated sra conductances by gsra,1 through gsra,N . We denote by wi,j and
Ei,j the synaptic weight and reversal potential at the synapse from cell j onto cell i. The
synaptic conductance that follows firing of cell j at time tj is the alpha function
αj (t) = (t − tj ) exp(3(tj − t)).
It follows then that the synaptic current onto cell i is
Isyn,i (t) =
N
X
wi,j αj (t)(Vi (t) − Ei,j )
j=1
So, we solve the full system
τm Vi0 (t) = EL − Vi (t) + Rm Ii (t) − rm gsra (t)(Vi (t) − EK ) − rm Isyn,i (t)
0
τsra gsra,i
(t) = −gsra,i (t)
We have coded this in iaf2net.m for the simple 7 cell 3 layer net below.
Rather than plotting spikes we simply track their times
iaf2net(10,−50,0.05,200)
8
7
6
7
6
cell
5
3
5
4
4
3
2
1
1
2
0
0
20
40
60
80
100
120
140
160
180
200
t (ms)
Figure 3. A net and its response. See iaf2net.m for details
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