mat2_psc_exp

mat2_psc_exp - Non-resetting leaky integrate-and-fire neuron model with exponential PSCs and adaptive threshold

Description

mat2_psc_exp is an implementation of a leaky integrate-and-fire model with exponential-kernel postsynaptic currents (PSCs). Thus, postsynaptic currents have an infinitely short rise time.

The threshold is lifted when the neuron is fired and then decreases in a fixed time scale toward a fixed level 3.

The threshold crossing is followed by a total refractory period during which the neuron is not allowed to fire, even if the membrane potential exceeds the threshold. The membrane potential is NOT reset, but continuously integrated.

Note

If tau_m is very close to tau_syn_exc or tau_syn_inh, 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

Rotter S and Diesmann M (1999). Exact simulation of time-invariant linear systems with applications to neuronal modeling. Biologial Cybernetics 81:381-402. DOI: https://doi.org/10.1007/s004220050570

2

Diesmann M, Gewaltig M-O, Rotter S, Aertsen A (2001). State space analysis of synchronous spiking in cortical neural networks. Neurocomputing 38-40:565-571. DOI:https://doi.org/10.1016/S0925-2312(01)00409-X

3

Kobayashi R, Tsubo Y and Shinomoto S (2009). Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold. Frontiers in Computuational Neuroscience 3:9. DOI: https://doi.org/10.3389/neuro.10.009.2009

Parameters

Name	Physical unit	Default value	Description
tau_m	ms	5ms	Membrane time constant
C_m	pF	100pF	Capacitance of the membrane
t_ref	ms	2ms	Duration of absolute refractory period (no spiking)
E_L	mV	-70mV	Resting potential
tau_syn_exc	ms	1ms	Time constant of postsynaptic excitatory currents
tau_syn_inh	ms	3ms	Time constant of postsynaptic inhibitory currents
tau_1	ms	10ms	Short time constant of adaptive threshold
tau_2	ms	200ms	Long time constant of adaptive threshold
alpha_1	mV	37mV	Amplitude of short time threshold adaption [3]
alpha_2	mV	2mV	Amplitude of long time threshold adaption [3]
omega	mV	19mV	Resting spike threshold (absolute value, not relative to E_L)
I_e	pA	0pA	constant external input current

State variables

Name	Physical unit	Default value	Description
V_th_alpha_1	mV	0mV	Two-timescale adaptive threshold
V_th_alpha_2	mV	0mV	Two-timescale adaptive threshold
r	integer	0	counts number of tick during the refractory period
V_abs	mV	0mV	Membrane potential
V_m	mV	V_abs + E_L	Relative 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 mat2_psc_exp:
  state:
    V_th_alpha_1 mV = 0mV # Two-timescale adaptive threshold
    V_th_alpha_2 mV = 0mV # Two-timescale adaptive threshold
    r integer = 0 # counts number of tick during the refractory period
    V_abs mV = 0mV # Membrane potential
    V_m mV = V_abs + E_L # Relative membrane potential.
    # I.e. the real threshold is (V_m-E_L).

  end
  equations:
    kernel I_kernel_inh = exp(-t / tau_syn_inh)
    kernel I_kernel_exc = exp(-t / tau_syn_exc)
    inline I_syn pA = convolve(I_kernel_exc,exc_spikes) - convolve(I_kernel_inh,inh_spikes)
    V_abs'=-V_abs / tau_m + (I_syn + I_e + I_stim) / C_m
  end

  parameters:
    tau_m ms = 5ms # Membrane time constant
    C_m pF = 100pF # Capacitance of the membrane
    t_ref ms = 2ms # Duration of absolute refractory period (no spiking)
    E_L mV = -70mV # Resting potential
    tau_syn_exc ms = 1ms # Time constant of postsynaptic excitatory currents
    tau_syn_inh ms = 3ms # Time constant of postsynaptic inhibitory currents
    tau_1 ms = 10ms # Short time constant of adaptive threshold
    tau_2 ms = 200ms # Long time constant of adaptive threshold
    alpha_1 mV = 37mV # Amplitude of short time threshold adaption [3]
    alpha_2 mV = 2mV # Amplitude of long time threshold adaption [3]
    omega mV = 19mV # Resting spike threshold (absolute value, not relative to E_L)
    # constant external input current

    # constant external input current
    I_e pA = 0pA
  end
  internals:
    h ms = resolution()
    P11th real = exp(-h / tau_1)
    P22th real = exp(-h / tau_2)
    RefractoryCounts integer = steps(t_ref) # refractory time in steps
  end
  input:
    exc_spikes pA <-excitatory spike
    inh_spikes pA <-inhibitory spike
    I_stim pA <-current
  end

  output: spike

  update:
    # evolve membrane potential
    integrate_odes()
    # evolve adaptive threshold
    V_th_alpha_1 = V_th_alpha_1 * P11th
    V_th_alpha_2 = V_th_alpha_2 * P22th
    if r == 0: # not refractory
      if V_abs >= omega + V_th_alpha_1 + V_th_alpha_2: # threshold crossing
        r = RefractoryCounts
        # procedure for adaptive potential
        V_th_alpha_1 += alpha_1 # short time
        V_th_alpha_2 += alpha_2 # long time
        emit_spike()
      end
    else:
      r = r - 1
    end
  end

end

Characterisation