Neuron

This chapter introduces how to choose neuron model and change some important parameters of the model.

neuron model

Neuron model is one of the most important component of the model. Different neuron model will have different neuron dynamics. In spiking neuron network, people always convert the change of membrane potential of neuron model into different equation and approximate it by difference equation. Finally, obtain the differential neuron model that can be computed by computer. In SPAIC , we contains most of the common neuron models:

• IF - Integrate-and-Fire model

• LIF - Leaky Integrate-and-Fire model

• CLIF - Current Leaky Integrate-and-Fire model

• GLIF - Generalized Leaky Integrate-and-Fire model

• aEIF - Adaptive Exponential Integrate-and-Fire model

• IZH - Izhikevich model

• HH - Hodgkin-Huxley model

In SPAIC , NeuronGroup is like nodes of the network model. Like layers in PyTorch , in SPAIC , NeuronGroup is the layer. Users need to specify the neuron numbers, neuron model or other related paramters.

from spaic import NeuronGroup

LIF neuron model

LIF(Leaky Integrated-and-Fire Model) neuron formula and parameters:

\[\begin{split}V & = tua\_m * V + I \\ O & = spike\_func(V^n)\end{split}\]

For example, we build a layer with 100 LIF neurons:

self.layer1 = NeuronGroup(num=100, model='lif')

A layer with 100 standard LIF neurons has been constructed. While, sometimes we need to specify the LIF neuron to get different neuron dynamics, that we will need to specify some parameters:

• tau_m - time constant of neuron membrane potential, default as 6.0

• v_th - the threshold voltage of a neuron, default as 1.0

• v_reset - the reset voltage of the neuron, which defaults to 0.0

If users need to change these parameters, they can enter the parameters when construct NeuronGroups .

self.layer2 = NeuronGroup(num=100, model='lif',
                tau_m=10.0, v_th=10, v_reset=0.2)

CLIF neuron model

CLIF(Current Leaky Integrated-and-Fire Model) neuron formula and parameters:

\[\begin{split}V(t) & = M(t) - S(t) - E(t) \\ I & = V0 * I \\ M & = tau\_p * M + I \\ S & = tau\_q * S + I \\ E & = tau\_p * E + Vth * O \\ O & = spike\_func(V)\end{split}\]

• tau_p, tau_q - time constants of synapse, default as 12.0 and 8.0

• tau_m - time constant of neuron membrane potential, default as 20.0

• v_th - the threshold voltage of a neuron, default as 1.0

GLIF neuron model

GLIF(Generalized Leaky Integrate-and-Fire Model) [1] neuron parameters:

• R, C, E_L

• Theta_inf

• f_v

• delta_v

• b_s

• delta_Theta_s

• k_1, k_2

• delta_I1, delta_I2

• a_v, b_v

aEIF neuron model

aEIF(Adaptive Exponential Integrated-and-Fire Model) [2] neuron model and parameters:

\[\begin{split}V & = V + dt / C * (gL * (EL - V + EXP) - w + I^n[t]) \\ w & = w + dt / tau\_w * (a * (V - EL) - w) \\ EXP & = delta\_t * exp(dv\_th/delta\_t) \\ dv & = V - EL \\ dv\_th & = V - Vth \\ O & = spike\_func(V^n) \\ if\quad V & > 20: \\ then\quad V & = EL, w = w + b\end{split}\]

• C, gL - membrane capacitance and leak conductance

• tau_w - adaptation time constant

• a. - subthreshold adaptation

• b. - spike-triggered adaptation

• delta_t - slope factor

• EL - leak reversal potential

IZH neuron model

IZH(Izhikevich Model) [3] neuron model and parameters:

\[\begin{split}V &= V + dt / tau\_M * (C1 * V * V + C2 * V + C3 - U + I) \\ V &= V + dt / tau\_M * (V* (C1 * V + C2) + C3 - U + I) \\ U &= U + a. * (b. * V - U) \\ O &= spike\_func(V^n) \\ if\quad V &> Vth, \\ then\quad V &= Vreset, U = U + d\end{split}\]

• tau_m

• C1, C2, C3

• a, b, d

• Vreset - Voltage Reset

HH neuron model

HH(Hodgkin-Huxley Model) [4] neuron model and parameters:

\[\begin{split}V & = V + dt/tau\_v * (I - Ik) \\ Ik & = NA + K + L \\ NA & = g\_NA * m^3 * h * (V - V_NA) \\ K & = g\_K * n^4 * (V - V_K) \\ L & = g\_L * (V - V_L) \\ K\quad activation: \\ n & = n + dt/tau\_n * (alpha\_n * (1-n) - beta\_n * n) \\ Na\quad activation: \\ m & = m + dt/tau\_m * (alpha\_m * (1-m) - beta\_m * m) \\ Na\quad inactivation: \\ h & = h + dt/tau\_h * (alpha\_h * (1-h) - beta\_h * h) \\ alpha\_m & = 0.1 * (-V + 25) / (exp((-V+25)/10) - 1) \\ beta\_m & = 4 * exp(-V/18) \\ alpha\_n & = 0.01 * (-V + 10) / (exp((-V+10)/10) - 1) \\ beta\_n & = 0.125 * exp(-V/80) \\ alpha\_h & = 0.07 * exp(-V/20) \\ beta\_h & = 1/(exp((-V+30)/10) + 1) \\ O & = spike\_func(V)\end{split}\]

• dt

• g_NA, g_K, g_L

• E_NA, E_K, E_L

• alpha_m1, alpha_m2, alpha_m3

• beta_m1, beta_m2, beta_m3

• alpha_n1, alpha_n2, alpha_n3

• beta_n1, beta_n2, beta_n3

• alpha_h1, alpha_h2, alpha_h3

• beta_1, beta_h2, beta_h3

• Vreset

• m, n, h

• V, v_th

customize

In the following chapter called Custom Neuron Model , we will talke about how to add custom neuron model into SPAIC with more details.

[1]

GLIF model : Teeter, C., Iyer, R., Menon, V., Gouwens, N., Feng, D., Berg, J., … & Mihalas, S. (2018). Generalized leaky integrate-and-fire models classify multiple neuron types. Nature communications, 9(1), 1-15.

[2]

AEIF model : Brette, Romain & Gerstner, Wulfram. (2005). Adaptive Exponential Integrate-And-Fire Model As An Effective Description Of Neuronal Activity. Journal of neurophysiology. 94. 3637-42.` doi:10.1152/jn.00686.2005. <https://doi.org/10.1152/jn.00686.2005>`_

[3]

IZH model : Izhikevich, E. M. (2003). Simple model of spiking neurons. IEEE Transactions on neural networks, 14(6), 1569-1572.

[4]

HH model : Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of physiology, 117(4), 500.