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Basic Models in Theoretical Neuroscience
Oren Shriki
2010
Synaptic Dynamics1
tt_spike
sg t
G
s
A spike at time tspike of the presynaptic cell contributes to the postsynaptic cell a time-dependent conductance , gs(t):
Synaptic Conductances
Peak conductance
2
Synaptic Dynamics: An Example• Synaptic dynamics are usually characterized by
fast rise and slow decay.
• The simplest model assumes instantaneous rise and exponential decay:
)(tGRg
dt
dg ss
spiket
spiketttR )( (Presynaptic rate)
3
Synaptic Dynamics: An Example• For a single presynaptic spike the solution is:
sts Getg )(
t
g s(t)
4
Synaptic Dynamics: An Example• Implementation in numerical simulations:
• Given the time step dt define the attenuation factor:
• A dimensionless parameter, f, is increased by 1 after each presynaptic spike and multiplied by the attenuation factor in each time step.
• The conductance is the product of f and the peak conductance, G.
sdtdt ee
5
Synaptic Dynamics• For simplicity, we shall write in general:
• K(t) is the time course (dimensionless) function.
• We define:
• For example:
)()( tGKtgs
0
)( dttKs
st
st ss edte
0
0 6
Network Architecture
External Inputs
Recurrent connectivity
1 2 3 4 N7
Voltage Dynamics for A Network of Conductance-Based Point Neurons
We assume that the neurons are point neurons obeying Hodgkin-Huxley type dynamics:
• Iactive – Ionic current involved in the action potential• Iext – External synaptic inputs• Inet – Synaptic inputs from within the network• Iapp – External current applied by the experimentalist
),,1( )( NiIIIIEtVgdt
dVC app
ineti
exti
activeiLiL
im
8
External Synaptic Current
• The explicit expression for the external synaptic current is:
• The peak synaptic conductance is:
• The time constant is:
inpiG
inpi
)()()( tVEtgtI iinpinp
iexti
9
Internal Synaptic Current
• The explicit expression for the internal synaptic current is:
• The peak synaptic conductance is:
• The time constant is:
ijG
ij
N
jijij
neti tVEtgtI
1
)()()(
10
