Learning in Spiking Neural Networks with a Calcium-based Hebbian Rule for Spike-timing-dependent Plasticity

TL;DR

The paper introduces a calcium-based Hebbian learning rule (BCaLL) for spiking neural networks that encodes pre- and post-synaptic activity with bounded calcium traces, enabling concurrent spike-timing and rate-based plasticity while remaining hardware-friendly. By integrating stop-learning, bistability, and coding-level-dependent inhibition, the authors demonstrate supervised MNIST classification with both feed-forward and recurrent architectures, and show that spike timing can modulate the learning rate without changing hyperparameters or mean firing rates. They further show that correlated spike timing—induced by subthreshold oscillations—significantly accelerates synaptic modification in recurrent networks, and that dynamics during continuous input presentations can reduce cross-class overlap in learned representations. Collectively, the work provides a mechanistic link between timing and rate in local synaptic plasticity, with implications for energy-efficient neuromorphic implementations and future exploration of diversity and attractor dynamics in SNNs.

Abstract

Understanding how biological neural networks are shaped via local plasticity mechanisms can lead to energy-efficient and self-adaptive information processing systems, which promises to mitigate some of the current roadblocks in edge computing systems. While biology makes use of spikes to seamless use both spike timing and mean firing rate to modulate synaptic strength, most models focus on one of the two. In this work, we present a Hebbian local learning rule that models synaptic modification as a function of calcium traces tracking neuronal activity. We show how the rule reproduces results from spike time and spike rate protocols from neuroscientific studies. Moreover, we use the model to train spiking neural networks on MNIST digit recognition to show and explain what sort of mechanisms are needed to learn real-world patterns. We show how our model is sensitive to correlated spiking activity and how this enables it to modulate the learning rate of the network without altering the mean firing rate of the neurons nor the hyparameters of the learning rule. To the best of our knowledge, this is the first work that showcases how spike timing and rate can be complementary in their role of shaping the connectivity of spiking neural networks.

Figures (27)

• Figure 1: Mean $x_{j}$ trace value as a function of firing rate for different $a_{i}$. With a trace jump of 1 the model restricts synaptic weights to the effects of only nearest-neighbor spike pairs (i.e. jump to maximum value). The colors show how the transition point between net weight decrease and net increase shift within the mean rate spectrum: notice how for $a_{i}=1$ the net $w_{hid}$ change goes from negative to positve within a very narrow range of the mean rate whereas for $a_{i}=0.1$ this range is much wider.
• Figure 2: One second simulation of a pair of neurons connected via a plastic synapse reproducing . The three main components of the rule are shown: the pre- ($x_{i}$, blue) and post-synaptic ($x_{j}$, red) calcium traces, updated at every spike, and the trace representing the weight hidden variable ($w_{hid}$, black). Each neuron is emitting Poisson spike trains at 20Hz. Hyperparameters in Table \ref{['tab:hyperparams_table']}.
• Figure 3: STDP (lef plot) and SRDP (right plot) curves generated by the BCaLL rule. STDP: the x-axis shows the time difference $\Delta t = t_{j} - t_{i}$ between pre- and post-synaptic spike times. The curve matches the depression for post-pre and potentiation for pre-post spike pairings; : change in synaptic weight as a function of spike pairing frequency with fixed $\Delta$t. Hyperparameters in Table \ref{['tab:hyperparams_table']}.
• Figure 4: Weight change as a function of mean rate pairing of pre and post neurons. While in the leftmost heat-map each neuron in the pair emits independent Poisson spike trains, in the middle and right most heat-maps positive and negative, respectively, time shifts are introduced in the spike pairs to bias them towards either post-pre or pre-post pairings. Hyperparameters in Table \ref{['tab:hyperparams_table']}.
• Figure 5: Test accuracy of a Linear Classifier trained on the MNIST dataset as a function of the gray-scale threshold applied to MNIST digits, for different number of units (pool size) allocated to encode each of the 10 output classes, and with floating point (float) or binary network parameters (bin): Mean (lines) and standard deviation (shaded area), over 5 independent parameter initializations. The dashed vertical line indicates the threshold value used in our simulations (i.e., 160).