wb_cond_exp

wb_cond_exp - Wang-Buzsaki model

Description

wb_cond_exp is an implementation of a modified Hodkin-Huxley model.

1. Post-synaptic currents: Incoming spike events induce a post-synaptic change of conductance modeled by an exponential function.

2. Spike Detection: Spike detection is done by a combined threshold-and-local- maximum search: if there is a local maximum above a certain threshold of the membrane potential, it is considered a spike.

References

1

Wang, X.J. and Buzsaki, G., (1996) Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. Journal of neuroscience, 16(20), pp.6402-6413.

See Also

hh_cond_exp_traub, wb_cond_multisyn

Parameters

Name	Physical unit	Default value	Description
t_ref	ms	2ms	Refractory period
g_Na	nS	3500nS	Sodium peak conductance
g_K	nS	900nS	Potassium peak conductance
g_L	nS	10nS	Leak conductance
C_m	pF	100pF	Membrane capacitance
E_Na	mV	55mV	Sodium reversal potential
E_K	mV	-90mV	Potassium reversal potential
E_L	mV	-65mV	Leak reversal potential (aka resting potential)
V_Tr	mV	-55mV	Spike threshold
tau_syn_exc	ms	0.2ms	Rise time of the excitatory synaptic alpha function
tau_syn_inh	ms	10ms	Rise time of the inhibitory synaptic alpha function
E_exc	mV	0mV	Excitatory synaptic reversal potential
E_inh	mV	-75mV	Inhibitory synaptic reversal potential
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	E_L	Membrane potential
Inact_h	real	alpha_h_init / (alpha_h_init + beta_h_init)
Act_n	real	alpha_n_init / (alpha_n_init + beta_n_init)

Equations

\[\frac{ dV_{m} } { dt }= \frac 1 { C_{m} } \left( { (-(I_{Na} + I_{K} + I_{L}) + I_{e} + I_{stim} + I_{syn,exc} - I_{syn,inh}) } \right)\]

\[\frac{ dAct_{n} } { dt }= (alpha_n(V_{m}) \cdot (1 - Act_{n}) - beta_n(V_{m}) \cdot Act_{n})\]

\[\frac{ dInact_{h} } { dt }= (alpha_h(V_{m}) \cdot (1 - Inact_{h}) - beta_h(V_{m}) \cdot Inact_{h})\]

Source code

neuron wb_cond_exp:
  state:
    r integer = 0 # number of steps in the current refractory phase
    V_m mV = E_L # Membrane potential
    Inact_h real = alpha_h_init / (alpha_h_init + beta_h_init)
    Act_n real = alpha_n_init / (alpha_n_init + beta_n_init)
  end
  equations:
    # synapses: exponential conductance
    kernel g_inh = exp(-t / tau_syn_inh)
    kernel g_exc = exp(-t / tau_syn_exc)
recordable    inline I_syn_exc pA = convolve(g_exc,exc_spikes) * (V_m - E_exc)
recordable    inline I_syn_inh pA = convolve(g_inh,inh_spikes) * (V_m - E_inh)
    inline I_Na pA = g_Na * _subexpr(V_m) * Inact_h * (V_m - E_Na)
    inline I_K pA = g_K * Act_n ** 4 * (V_m - E_K)
    inline I_L pA = g_L * (V_m - E_L)
    V_m'=(-(I_Na + I_K + I_L) + I_e + I_stim + I_syn_exc - I_syn_inh) / C_m
    Act_n'=(alpha_n(V_m) * (1 - Act_n) - beta_n(V_m) * Act_n) # n-variable
    Inact_h'=(alpha_h(V_m) * (1 - Inact_h) - beta_h(V_m) * Inact_h) # h-variable
  end

  parameters:
    t_ref ms = 2ms # Refractory period
    g_Na nS = 3500nS # Sodium peak conductance
    g_K nS = 900nS # Potassium peak conductance
    g_L nS = 10nS # Leak conductance
    C_m pF = 100pF # Membrane capacitance
    E_Na mV = 55mV # Sodium reversal potential
    E_K mV = -90mV # Potassium reversal potential
    E_L mV = -65mV # Leak reversal potential (aka resting potential)
    V_Tr mV = -55mV # Spike threshold
    tau_syn_exc ms = 0.2ms # Rise time of the excitatory synaptic alpha function
    tau_syn_inh ms = 10ms # Rise time of the inhibitory synaptic alpha function
    E_exc mV = 0mV # Excitatory synaptic reversal potential
    E_inh mV = -75mV # Inhibitory synaptic reversal potential
    # constant external input current

    # constant external input current
    I_e pA = 0pA
  end
  internals:
    RefractoryCounts integer = steps(t_ref) # refractory time in steps
    alpha_n_init 1/ms = -0.05 / (ms * mV) * (E_L + 34.0mV) / (exp(-0.1 * (E_L + 34.0mV)) - 1.0)
    beta_n_init 1/ms = 0.625 / ms * exp(-(E_L + 44.0mV) / 80.0mV)
    alpha_h_init 1/ms = 0.35 / ms * exp(-(E_L + 58.0mV) / 20.0mV)
    beta_h_init 1/ms = 5.0 / (exp(-0.1 / mV * (E_L + 28.0mV)) + 1.0) / ms
  end
  input:
    inh_spikes nS <-inhibitory spike
    exc_spikes nS <-excitatory spike
    I_stim pA <-current
  end

  output: spike

  update:
    U_old mV = V_m
    integrate_odes()
    # sending spikes: crossing 0 mV, pseudo-refractoriness and local maximum...

    # sending spikes: crossing 0 mV, pseudo-refractoriness and local maximum...
    if r > 0: # is refractory?
      r -= 1
    elif V_m > V_Tr and U_old > V_m:
      r = RefractoryCounts
      emit_spike()
    end
  end

function _subexpr(V_m mV) real:
    return alpha_m(V_m) ** 3 / (alpha_m(V_m) + beta_m(V_m)) ** 3
end

function alpha_m(V_m mV) 1/ms:
    return 0.1 / (ms * mV) * (V_m + 35.0mV) / (1.0 - exp(-0.1mV * (V_m + 35.0mV)))
end

function beta_m(V_m mV) 1/ms:
    return 4.0 / (ms) * exp(-(V_m + 60.0mV) / 18.0mV)
end

function alpha_n(V_m mV) 1/ms:
    return -0.05 / (ms * mV) * (V_m + 34.0mV) / (exp(-0.1 * (V_m + 34.0mV)) - 1.0)
end

function beta_n(V_m mV) 1/ms:
    return 0.625 / ms * exp(-(V_m + 44.0mV) / 80.0mV)
end

function alpha_h(V_m mV) 1/ms:
    return 0.35 / ms * exp(-(V_m + 58.0mV) / 20.0mV)
end

function beta_h(V_m mV) 1/ms:
    return 5.0 / (exp(-0.1 / mV * (V_m + 28.0mV)) + 1.0) / ms
end

end

Characterisation