izhikevich_psc_alpha

Parameters

Name	Physical unit	Default value	Description
C_m	pF	200pF	Membrane capacitance
k	pF / (ms mV)	8pF / mV / ms	Spiking slope
V_r	mV	-65mV	Resting potential
V_t	mV	-45mV	Threshold potential
a	1 / ms	0.01 / ms	Time scale of recovery variable
b	nS	9nS	Sensitivity of recovery variable
c	mV	-65mV	After-spike reset value of V_m
d	pA	60pA	After-spike reset value of U_m
V_peak	mV	0mV	Spike detection threshold (reset condition)
tau_syn_exc	ms	0.2ms	Synaptic time constant of excitatory synapse
tau_syn_inh	ms	2ms	Synaptic time constant of inhibitory synapse
t_ref	ms	2ms	Refractory period
I_e	pA	0pA	constant external input current

State variables

Name	Physical unit	Default value	Description
r	integer	0	Number of steps in the current refractory phase
V_m	mV	-65mV	Membrane potential
U_m	pA	0pA	Membrane potential recovery variable

Equations

\[\frac{ dV_{m} } { dt }= \frac 1 { C_{m} } \left( { (k \cdot (V_{m} - V_{r}) \cdot (V_{m} - V_{t}) - U_{m} + I_{e} + I_{stim} + I_{syn,exc} - I_{syn,inh}) } \right)\]

\[\frac{ dU_{m} } { dt }= a \cdot (b \cdot (V_{m} - V_{r}) - U_{m})\]

Source code

neuron izhikevich_psc_alpha:
  state:
    r integer = 0 # Number of steps in the current refractory phase
    V_m mV = -65mV # Membrane potential
    U_m pA = 0pA # Membrane potential recovery variable
  end
  equations:
    # synapses: alpha functions
    kernel K_syn_inh = (e / tau_syn_inh) * t * exp(-t / tau_syn_inh)
    kernel K_syn_exc = (e / tau_syn_exc) * t * exp(-t / tau_syn_exc)
    inline I_syn_exc pA = convolve(K_syn_exc,exc_spikes)
    inline I_syn_inh pA = convolve(K_syn_inh,inh_spikes)
    V_m'=(k * (V_m - V_r) * (V_m - V_t) - U_m + I_e + I_stim + I_syn_exc - I_syn_inh) / C_m
    U_m'=a * (b * (V_m - V_r) - U_m)
  end

  parameters:
    C_m pF = 200pF # Membrane capacitance
    k pF/mV/ms = 8pF / mV / ms # Spiking slope
    V_r mV = -65mV # Resting potential
    V_t mV = -45mV # Threshold potential
    a 1/ms = 0.01 / ms # Time scale of recovery variable
    b nS = 9nS # Sensitivity of recovery variable
    c mV = -65mV # After-spike reset value of V_m
    d pA = 60pA # After-spike reset value of U_m
    V_peak mV = 0mV # Spike detection threshold (reset condition)
    tau_syn_exc ms = 0.2ms # Synaptic time constant of excitatory synapse
    tau_syn_inh ms = 2ms # Synaptic time constant of inhibitory synapse
    t_ref ms = 2ms # Refractory period
    # constant external input current

    # constant external input current
    I_e pA = 0pA
  end
  internals:
    RefractoryCounts integer = steps(t_ref) # refractory time in steps
  end
  input:
    inh_spikes pA <-inhibitory spike
    exc_spikes pA <-excitatory spike
    I_stim pA <-current
  end

  output: spike

  update:
    integrate_odes()
    # refractoriness and threshold crossing
    if r > 0: # is refractory?
      r -= 1
    elif V_m >= V_peak:
      V_m = c
      U_m += d
      emit_spike()
      r = RefractoryCounts
    end
  end

end

Characterisation