iaf_psc_exp

iaf_psc_exp - Leaky integrate-and-fire neuron model with exponential PSCs

Description

iaf_psc_exp is an implementation of a leaky integrate-and-fire model with exponential-kernel postsynaptic currents (PSCs) according to 1. Thus, postsynaptic currents have an infinitely short rise time.

The threshold crossing is followed by an absolute refractory period (t_ref) during which the membrane potential is clamped to the resting potential and spiking is prohibited.

Note

If tau_m is very close to tau_syn_ex or tau_syn_in, numerical problems may arise due to singularities in the propagator matrics. If this is the case, replace equal-valued parameters by a single parameter.

For details, please see IAF_neurons_singularity.ipynb in the NEST source code (docs/model_details).

References

1

Tsodyks M, Uziel A, Markram H (2000). Synchrony generation in recurrent networks with frequency-dependent synapses. The Journal of Neuroscience, 20,RC50:1-5. URL: https://infoscience.epfl.ch/record/183402

See also

iaf_cond_exp

Author

Moritz Helias

Parameters

Name	Physical unit	Default value	Description
C_m	pF	250pF	Capacity of the membrane
tau_m	ms	10ms	Membrane time constant.
tau_syn_in	ms	2ms	Time constant of synaptic current.
tau_syn_ex	ms	2ms	Time constant of synaptic current.
t_ref	ms	2ms	Duration of refractory period
E_L	mV	-70mV	Resting potential.
V_reset	mV	-70mV - E_L	reset value of the membrane potential
Theta	mV	-55mV - E_L	Threshold, RELATIVE TO RESTING POTENTIAL(!).
I_e	pA	0pA	constant external input current

State variables

Name	Physical unit	Default value	Description
V_abs	mV	0mV
V_m	mV	V_abs + E_L	Membrane potential.

Equations

\[\frac{ dV_{abs} } { dt }= \frac{ -V_{abs} } { \tau_{m} } + \frac 1 { C_{m} } \left( { (I_{syn} + I_{e} + I_{stim}) } \right)\]

Source code

neuron iaf_psc_exp:
   state:
     r integer = 0  # counts number of tick during the refractory period
     V_abs mV = 0 mV
   end

   equations:
     kernel I_kernel_in = exp(-1/tau_syn_in*t)
     kernel I_kernel_ex = exp(-1/tau_syn_ex*t)
     recordable inline V_m mV = V_abs + E_L # Membrane potential.
     inline I_syn pA = convolve(I_kernel_in, in_spikes) + convolve(I_kernel_ex, ex_spikes) + I_e + I_stim
     V_abs' = -V_abs / tau_m + I_syn / C_m
   end

   parameters:
     C_m pF = 250 pF       # Capacity of the membrane
     tau_m ms = 10 ms      # Membrane time constant
     tau_syn_in ms = 2 ms  # Time constant of synaptic current
     tau_syn_ex ms = 2 ms  # Time constant of synaptic current
     t_ref ms = 2 ms       # Duration of refractory period
     E_L  mV = -70 mV      # Resting potential
     V_reset mV = -70 mV - E_L # reset value of the membrane potential
     Theta   mV = -55 mV - E_L # Threshold, RELATIVE TO RESTING POTENTIAL (!).
                                    # I.e. the real threshold is (E_L_+V_th_)

     # constant external input current
     I_e pA = 0 pA
   end

   internals:
     RefractoryCounts integer = steps(t_ref) # refractory time in steps
   end

   input:
     ex_spikes pA <- excitatory spike
     in_spikes pA <- inhibitory spike
     I_stim pA <- continuous
   end

   output: spike

   update:
     if r == 0: # neuron not refractory, so evolve V
       integrate_odes()
     else:
       r = r - 1 # neuron is absolute refractory
     end

     if V_abs >= Theta: # threshold crossing
       r = RefractoryCounts
       V_abs = V_reset
       emit_spike()
     end

   end

 end

Characterisation

Synaptic response

f-I curve