hh_psc_alpha¶

hh_psc_alpha - Hodgkin-Huxley neuron model

Description¶

hh_psc_alpha is an implementation of a spiking neuron using the Hodgkin-Huxley formalism.

Incoming spike events induce a post-synaptic change of current modelled by an alpha function. The alpha function is normalised such that an event of weight 1.0 results in a peak current of 1 pA.

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.

Problems/Todo¶

better spike detection
initial wavelet/spike at simulation onset

References¶

1: Gerstner W, Kistler W (2002). Spiking neuron models: Single neurons, populations, plasticity. New York: Cambridge University Press
2: Dayan P, Abbott LF (2001). Theoretical neuroscience: Computational and mathematical modeling of neural systems. Cambridge, MA: MIT Press. https://pure.mpg.de/pubman/faces/ViewItemOverviewPage.jsp?itemId=item_3006127>
3: Hodgkin AL and Huxley A F (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology 117. DOI: https://doi.org/10.1113/jphysiol.1952.sp004764

Parameters¶

Name	Physical unit	Default value	Description
t_ref	ms	2.0ms	Refractory period
g_Na	nS	12000.0nS	Sodium peak conductance
g_K	nS	3600.0nS	Potassium peak conductance
g_L	nS	30nS	Leak conductance
C_m	pF	100.0pF	Membrane Capacitance
E_Na	mV	50mV	Sodium reversal potential
E_K	mV	-77.0mV	Potassium reversal potentia
E_L	mV	-54.402mV	Leak reversal Potential (aka resting potential)
tau_syn_ex	ms	0.2ms	Rise time of the excitatory synaptic alpha function i
tau_syn_in	ms	2.0ms	Rise time of the inhibitory synaptic alpha function
I_e	pA	0pA	constant external input current

State variables¶

Name	Physical unit	Default value	Description
V_m	mV	-65.0mV	Membrane potential
alpha_n_init	real	(0.01 * (V_m / mV + 55.0)) / (1.0 - exp(-(V_m / mV + 55.0) / 10.0))
beta_n_init	real	0.125 * exp(-(V_m / mV + 65.0) / 80.0)
alpha_m_init	real	(0.1 * (V_m / mV + 40.0)) / (1.0 - exp(-(V_m / mV + 40.0) / 10.0))
beta_m_init	real	4.0 * exp(-(V_m / mV + 65.0) / 18.0)
alpha_h_init	real	0.07 * exp(-(V_m / mV + 65.0) / 20.0)
beta_h_init	real	1.0 / (1.0 + exp(-(V_m / mV + 35.0) / 10.0))
Act_m	real	alpha_m_init / (alpha_m_init + beta_m_init)	Activation variable m for Na
Inact_h	real	alpha_h_init / (alpha_h_init + beta_h_init)	Inactivation variable h for Na
Act_n	real	alpha_n_init / (alpha_n_init + beta_n_init)	Activation variable n for K

Equations¶

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

\[\frac{ dAct_{n} } { dt }= \frac 1 { \mathrm{ms} } \left( { (\alpha_{n} \cdot (1 - Act_{n}) - \beta_{n} \cdot Act_{n}) } \right)\]

\[\frac{ dAct_{m} } { dt }= \frac 1 { \mathrm{ms} } \left( { (\alpha_{m} \cdot (1 - Act_{m}) - \beta_{m} \cdot Act_{m}) } \right)\]

\[\frac{ dInact_{h} } { dt }= \frac 1 { \mathrm{ms} } \left( { (\alpha_{h} \cdot (1 - Inact_{h}) - \beta_{h} \cdot Inact_{h}) } \right)\]

Source code¶neuron hh_psc_alpha:
  state:
    r integer = 0 # number of steps in the current refractory phase

    V_m mV = -65.0mV # Membrane potential
    Act_m real = alpha_m_init / (alpha_m_init + beta_m_init) # Activation variable m for Na
    Inact_h real = alpha_h_init / (alpha_h_init + beta_h_init) # Inactivation variable h for Na
    Act_n real = alpha_n_init / (alpha_n_init + beta_n_init) # Activation variable n for K
  end

  equations:
    # synapses: alpha functions
    kernel I_syn_in = (e / tau_syn_in) * t * exp(-t / tau_syn_in)
    kernel I_syn_ex = (e / tau_syn_ex) * t * exp(-t / tau_syn_ex)
    inline I_syn_exc pA = convolve(I_syn_ex,spikeExc)
    inline I_syn_inh pA = convolve(I_syn_in,spikeInh)
    inline I_Na pA = g_Na * Act_m * Act_m * Act_m * Inact_h * (V_m - E_Na)
    inline I_K pA = g_K * Act_n * Act_n * Act_n * Act_n * (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_inh + I_syn_exc) / C_m

    # Act_n
    inline alpha_n real = (0.01 * (V_m / mV + 55.0)) / (1.0 - exp(-(V_m / mV + 55.0) / 10.0))
    inline beta_n real = 0.125 * exp(-(V_m / mV + 65.0) / 80.0)
    Act_n'=(alpha_n * (1 - Act_n) - beta_n * Act_n) / ms # n-variable

    # Act_m
    inline alpha_m real = (0.1 * (V_m / mV + 40.0)) / (1.0 - exp(-(V_m / mV + 40.0) / 10.0))
    inline beta_m real = 4.0 * exp(-(V_m / mV + 65.0) / 18.0)
    Act_m'=(alpha_m * (1 - Act_m) - beta_m * Act_m) / ms # m-variable

    # Inact_h'
    inline alpha_h real = 0.07 * exp(-(V_m / mV + 65.0) / 20.0)
    inline beta_h real = 1.0 / (1.0 + exp(-(V_m / mV + 35.0) / 10.0))
    Inact_h'=(alpha_h * (1 - Inact_h) - beta_h * Inact_h) / ms # h-variable
  end

  parameters:
    t_ref ms = 2.0ms # Refractory period
    g_Na nS = 12000.0nS # Sodium peak conductance
    g_K nS = 3600.0nS # Potassium peak conductance
    g_L nS = 30nS # Leak conductance
    C_m pF = 100.0pF # Membrane Capacitance
    E_Na mV = 50mV # Sodium reversal potential
    E_K mV = -77.0mV # Potassium reversal potentia
    E_L mV = -54.402mV # Leak reversal Potential (aka resting potential)
    tau_syn_ex ms = 0.2ms # Rise time of the excitatory synaptic alpha function i
    tau_syn_in ms = 2.0ms # Rise time of the inhibitory synaptic alpha function

    # constant external input current
    I_e pA = 0pA
  end

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

  input:
    spikeInh pA <-inhibitory spike
    spikeExc pA <-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...
    if r > 0: # is refractory?
      r -= 1
    elif V_m > 0mV and U_old > V_m:
      r = RefractoryCounts
      emit_spike()
    end
  end

end

Characterisation¶

Synaptic response¶