Neural model specifications

This page gives the mathematical specification of the neuron, and learning-rule models BindsNET implements: the difference equations actually solved each timestep and the default parameters with units. Equations and defaults below were transcribed from the source (bindsnet/network/nodes.py and bindsnet/learning/learning.py); when in doubt, the source is authoritative. Parameter defaults are stated as of package version 0.3.4.

Discretization

BindsNET does not integrate ODEs symbolically. Continuous-time dynamics are converted to difference equations and advanced at a fixed timestep \(dt\) (milliseconds; the examples use \(dt = 1.0\)). A network-wide \(dt\) is set on the Network object. Exponential leak terms are precomputed once per \(dt\) as a decay factor

\[\text{decay} = \exp\!\left(-\,dt / \tau\right),\]

so a leaky variable \(y\) relaxing toward a baseline \(y_0\) updates as \(y \leftarrow \text{decay}\,(y - y_0) + y_0\).

Notation: \(v\) membrane voltage, \(v_\text{rest}\) rest, \(v_\text{reset}\) post-spike reset, \(v_\text{thr}\) threshold, \(s\) spike (boolean), \(x\) spike trace, \(\tau\) a time constant. All voltages are in millivolts and follow the biological convention used in the code (e.g. rest \(-65\)mV).

Neuron models

All neuron layers live in bindsnet.network.nodes. Spikes are emitted when the (possibly adapted) threshold is crossed; most models then apply a reset and a refractory period refrac during which inputs are ignored. Optional spike traces decay with time constant tc_trace (default 20 ms) and are used by the trace-based learning rules.

McCulloch–Pitts (McCullochPitts)

Stateless threshold unit; the voltage equals the input and a spike is emitted when it reaches threshold.

\[v_t = x_t, \qquad s_t = \big[\,v_t \ge v_\text{thr}\,\big]\]

Defaults: thresh \(= 1.0\).

Integrate-and-fire (IFNodes)

Non-leaky accumulator with reset and refractory period.

\[v_t = v_{t-1} + \mathbb{1}[\text{refrac}\le 0]\,x_t, \qquad s_t = [\,v_t \ge v_\text{thr}\,], \qquad v_t \leftarrow v_\text{reset}\ \text{if}\ s_t\]

Defaults: thresh \(=-52\), reset \(=-65\), refrac \(=5\).

Leaky integrate-and-fire (LIFNodes)

Leak toward rest, integrate input, threshold-reset-refractory.

\[v_t = \text{decay}\,(v_{t-1} - v_\text{rest}) + v_\text{rest} + x_t, \qquad \text{decay} = \exp(-dt/\tau_\text{decay})\]

\[s_t = [\,v_t \ge v_\text{thr}\,], \qquad v_t \leftarrow v_\text{reset}\ \text{if}\ s_t\]

Defaults: thresh \(=-52\), rest \(=-65\), reset \(=-65\), refrac \(=5\), tc_decay \(=100\)ms. Inputs are masked to zero while a neuron is refractory.

BoostedLIFNodes is a performance-oriented LIF variant (per source: no separate rest/reset/lower-bound handling); use it when those features are not needed.

Current-based LIF (CurrentLIFNodes)

Adds a decaying synaptic current \(i\) between input and membrane:

\[i_t = i_\text{decay}\,i_{t-1} + x_t, \qquad v_t = \text{decay}\,(v_{t-1} - v_\text{rest}) + v_\text{rest} + \mathbb{1}[\text{refrac}\le 0]\,i_t\]

with \(i_\text{decay} = \exp(-dt/\tau_{i})\). See source for the tc_i_decay default and the remaining (LIF-shared) parameters.

Adaptive-threshold LIF (AdaptiveLIFNodes)

LIF with a threshold adaptation variable \(\theta\) that increases on each spike and decays otherwise:

\[\theta_t = \theta_\text{decay}\,\theta_{t-1} + \theta_+ \textstyle\sum s_t, \qquad s_t = [\,v_t \ge v_\text{thr} + \theta_t\,]\]

Defaults add theta_plus \(=0.05\), tc_theta_decay \(=10^{7}\)ms (on top of the LIF defaults). Adaptation is applied while learning is enabled.

Diehl & Cook 2015 (DiehlAndCookNodes)

Adaptive-threshold LIF tuned for the Diehl & Cook (2015) MNIST replication, with the additional one_spike option (default True) that permits at most one spike per layer per timestep. Same parameter defaults as AdaptiveLIFNodes. Used by the DiehlAndCook2015 model and examples/mnist/eth_mnist.py.

Izhikevich (IzhikevichNodes)

Two-variable model (membrane \(v\), recovery \(u\)) integrated with two half Euler steps per timestep:

\[v \leftarrow v + \tfrac{dt}{2}\,(0.04 v^2 + 5 v + 140 - u + x)\quad(\text{applied twice}), \qquad u \leftarrow u + dt\,a\,(b v - u)\]

On spike (\(v \ge v_\text{thr}\)): \(v \leftarrow c\), \(u \leftarrow u + d\). Excitatory/inhibitory populations are parameterized as in Izhikevich (2003): excitatory \(a=0.02,\,b=0.2,\,c=-65+15r^2,\,d=8-6r^2\); inhibitory \(a=0.02+0.08r,\,b=0.25-0.05r,\,c=-65,\,d=2\), with \(r\sim U(0,1)\).

SRM0 (SRM0Nodes)

Simplified Spike Response Model with stochastic (“escape noise”) firing:

\[v_t = \text{decay}\,(v_{t-1}-v_\text{rest}) + v_\text{rest} + \mathbb{1}[\text{refrac}\le 0]\,\varepsilon_0\,x_t\]

\[\rho = \rho_0 \exp\!\Big(\tfrac{v_t - v_\text{thr}}{\Delta_\text{thr}}\Big), \qquad P(\text{spike}) = 1 - e^{-\rho\,dt}, \qquad s_t = [\,U(0,1) < P(\text{spike})\,]\]

Cumulative SRM (CSRMNodes)

Cumulative Spike Response Model (Gerstner & van Hemmen 1992; Gerstner et al. 1996): refractoriness and adaptation arise from the summed after-potentials of several previous spikes rather than only the most recent one. See the source for the response-kernel implementation.

Input (Input)

Passes externally provided spike tensors (e.g. from bindsnet.encoding) into the network; it has no internal membrane dynamics.

Learning rules

Learning rules live in bindsnet.learning. They modify connection weights w from pre-synaptic spikes/traces (source) and post-synaptic spikes/traces (target). nu is the (pre, post) learning-rate pair; reduction aggregates over the batch; weight_decay optionally decays weights each step.

Post-pre STDP (PostPre)

Trace-based spike-timing-dependent plasticity (requires traces on both layers). For a dense Connection the per-step update is

\[\Delta w = -\,\nu_\text{pre}\;(s_\text{pre} \otimes x_\text{post}) \;+\; \nu_\text{post}\;(x_\text{pre} \otimes s_\text{post})\]

i.e. a pre-synaptic spike depresses the synapse in proportion to the post-synaptic trace, and a post-synaptic spike potentiates it in proportion to the pre-synaptic trace. Convolutional and locally-connected variants apply the same rule patch-wise.

Hebbian (Hebbian)

Both pre- and post-synaptic events increase the weight (no depression term), proportional to the opposite layer’s trace.

Weight-dependent post-pre (WeightDependentPostPre)

PostPre whose potentiation/depression magnitudes are scaled by the distance of the weight from its bounds (wmin/wmax), yielding soft saturation at the limits.

Reward-modulated STDP (MSTDP, MSTDPET)

Three-factor rules: a STDP-like eligibility signal is gated by a scalar reward. MSTDP modulates the immediate pre/post correlation by reward; MSTDPET adds an eligibility trace that accumulates the correlation over time (time constant tc_e_trace) before reward gating. Reward is supplied via the pipeline / an AbstractReward (e.g. MovingAvgRPE). See source for the exact eligibility update.

Rmax (Rmax)

Reward-maximizing rule intended for stochastic (SRM0) neurons; see source for its formulation.

Note

Where this page summarizes a rule “see source”, the equations were not reproduced here to avoid mis-stating constants; consult bindsnet/learning/learning.py for the authoritative form. If an implementation deviates from a textbook model, the code is the specification.