Quantization-based simulation of spiking neurons: theoretical properties and performance analysis

ABSTRACT

In this work we present an exhaustive analysis of the use of Quantized State Systems (QSS) algorithms for the discrete event simulation of Leaky Integrate and Fire models of spiking neurons. Making use of some properties of these algorithms, we first derive theoretical error bounds for the sub-threshold dynamics as well as estimates of the computational costs as a function of the accuracy settings. Then, we corroborate those results on different simulation experiments, where we also study how these algorithms scale with the size of the network and its connectivity. The results obtained show that the QSS algorithms, without any type of optimisation or specialisation, obtain accurate results with low computational costs even in large networks with a high level of connectivity.

KEYWORDS:

• Hybrid systems
• discontinuity handling
• quantized state systems
• spiking neural networks
• event-driven simulation

Acknowledgments

This work was supported by ANPCYT under Grant PICT–2017 2436, CONICET under Grant PIP 2022-2024 11220210100093CO and CNRS PRIME eXplAIn research team.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. The expression $|\mathbf{M}|$ denotes the component-wise absolute value of a vector or matrix $\mathbf{M}$. Also, $\mathbf{a}\mathbf{b}$ expresses a set of component-wise inequalities $a_i\le b_i$ on all the components of $\mathbf{a}\in {\mathbb{R}}^n$ and $\mathbf{b}\in {\mathbb{R}}^n$.

2. We did not include Linearly Implicit QSS Methods because the models are not stiff.

3. The QSS Solvers uses the standard stdlib C library for generating pseudo-random sequences.

4. The quantum was expressed in mV and nA in the different tables, but the model considers that the membrane potential and the synaptic current are measured in SI units (V and A, respectively).

5. The expected number of steps is the result obtained from Eq. (31)(31) $k_{x_i(t_0.t_f)}^{(n)}(\mathrm{\Delta }{Q}_i)\approx \frac{A_{0,T_s}^{(n)}}{{(\mathrm{\Delta }{Q}_i)}^{1/n}}=n\frac{I_m^{1/n}-I_0^{1/n}}{{(n!\mathrm{\Delta }{Q}_i)}^{1/n}}$(31) plus one step corresponding to the arrival of the input spike.

Additional information

Funding

The work was supported by the Agencia Nacional de Promoción Científica y Tecnológica [PICT–2017 2436]; Consejo Nacional de Investigaciones Científicas y Técnicas [PIP 2022-2024 11220210100093CO]; CNRS PRIME eXplAIn research team .