LIF

Here you find tools for the leaky integrate-and-fire (LIF) neuron. The module is divided into submodules corresponding to synapse types (see theory section below).

nnmt.lif.delta Collection of functions for LIF neurons with delta synapses.
nnmt.lif.exp Collection of functions for LIF neurons with exponential synapses.
nnmt.lif._general Collection of functions used by both lif.delta and lif.exp.

Theory

The differential equation describing the membrane potential V of the leaky integrate-and-fire (LIF) neuron is given by

\tau_\mathrm{m} \frac{\mathrm{d}V}{\mathrm{d}t} = -V + I_{\mathrm{syn}}(V,t) + I_\mathrm{ext}(t) \quad ,

with membrane time constant \tau_\mathrm{m}, synaptic current I_{\mathrm{syn}}(V,t), and external current I_{\mathrm{ext}}(t). Once the membrane voltage reaches a threshold V_\Theta, it is reset to the reset potential V_0 and a spike is emitted.

See Fourcaud and Brunel [2002] for more details.

Delta synapses

For instantaneous or delta synapses the synaptic current is given by

I_{\mathrm{syn}}(t) = \sum_{i=1}^{N_\mathrm{s}} J_i \sum_k \delta(t-t_i^k) \tau_\mathrm{m} \quad ,

where the first sum runs over all N_\mathrm{s} synapses and the second sum over all presynaptic spikes of each synapse. t_i^k is the time at which spike k arrives at synapse i. J_i is the synaptic efficacy or weight of synapse i.

Exponential synapses

For synapses with instantaneous jump and exponential decay with time constant \tau_\mathrm{s}, or just exponential synapses, the synaptic current is given by

\tau_\mathrm{s}\frac{\mathrm{d} I_{\mathrm{syn}}}{\mathrm{d} t} = -I_\mathrm{syn}(t) + \sum_{i=1}^{N_\mathrm{s}} J_i \sum_k \delta(t-t_i^k) \tau_\mathrm{m} \quad .

Variables

Here you find how variables of LIF neurons are named in NNMT:

Quantity Symbol Variable name
Synaptic delay matrix \boldsymbol{D} D
Synaptic weight matrix \boldsymbol{J} J
External synaptic weight matrix \boldsymbol{J}_\mathrm{ext} J_ext
Indegree matrix \boldsymbol{K} K
External indegree matrix \boldsymbol{K}_\mathrm{ext} K_ext
Number of neurons \boldsymbol{N} N
Reset voltage V_0 V_0
Threshold voltage V_\Theta V_th
Mean of synaptic input \boldsymbol{\mu} mu
Population rates \boldsymbol{\nu} nu
External population rates \boldsymbol{\nu}_\mathrm{ext} nu_ext
Angular frequencies \omega omegas
Standard deviation of synaptic input \boldsymbol{\sigma} sigma
Post-synaptic membrane time constant \tau_\mathrm{m} tau_m
Refractory time \tau_\mathrm{r} tau_r
Pre-synaptic time constant \tau_\mathrm{s} tau_s