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_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: 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

Author¶

Thomas Pfeil (modified iaf_psc_exp model of Moritz Helias)

Parameters¶

Name	Physical unit	Default value	Description
tau_m	ms	5ms	Membrane time constant
C_m	pF	100pF	Capacity of the membrane
t_ref	ms	2ms	Duration of absolute refractory period (no spiking)
E_L	mV	-70.0mV	Resting potential
tau_syn_ex	ms	1ms	Time constant of postsynaptic excitatory currents
tau_syn_in	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	37.0mV	Amplitude of short time threshold adaption [3]
alpha_2	mV	2.0mV	Amplitude of long time threshold adaption [3]
omega	mV	19.0mV	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_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 = 0 mV # Two-timescale adaptive threshold
    V_th_alpha_2 mV = 0 mV # Two-timescale adaptive threshold

    r integer = 0 # counts number of tick during the refractory period
    V_abs mV  = 0 mV # 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_in = exp(-1/tau_syn_in*t)
    kernel I_kernel_ex = exp(-1/tau_syn_ex*t)

    inline I_syn pA = convolve(I_kernel_in, in_spikes) + convolve(I_kernel_ex, ex_spikes)
    V_abs' = -V_abs / tau_m + (I_syn + I_e + I_stim) / C_m
  end

  parameters:
    tau_m        ms =     5 ms  # Membrane time constant
    C_m          pF =   100 pF  # Capacity of the membrane
    t_ref        ms =     2 ms  # Duration of absolute refractory period (no spiking)
    E_L          mV = -70.0 mV  # Resting potential
    tau_syn_ex   ms =     1 ms  # Time constant of postsynaptic excitatory currents
    tau_syn_in   ms =     3 ms  # Time constant of postsynaptic inhibitory currents
    tau_1        ms =    10 ms  # Short time constant of adaptive threshold
    tau_2        ms =   200 ms  # Long time constant of adaptive threshold
    alpha_1      mV =  37.0 mV  # Amplitude of short time threshold adaption [3]
    alpha_2      mV =   2.0 mV  # Amplitude of long time threshold adaption [3]
    omega        mV =  19.0 mV  # Resting spike threshold (absolute value, not relative to E_L)

    # constant external input current
    I_e pA = 0 pA
  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:
    ex_spikes pA <- excitatory spike
    in_spikes pA <- inhibitory spike
    I_stim pA <- continuous
  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