https://doi.org/10.1140/epjs/s11734-025-02117-6
Regular Article
Computing fractional-order neural responses through multiscale relaxation dynamics
1
Saint-Petersburg State Research Institute of Phthisiopulmonology, Ligovskiy Av. 2-4, 194064, Saint Petersburg, Russia
2
Sophya Kovalevskaya North-West Mathematical Research Center, Immanuel Kant Baltic Federal University, Nevskogo St. 14, 236041, Kaliningrad, Russia
3
Theoretical Physics Department, Kursk State University, Radishcheva St., 33, 305000, Kursk, Russia
Received:
12
November
2025
Accepted:
15
December
2025
Published online:
24
December
2025
Experimental studies show that neuronal membrane responses often decay non-uniformly across multiple time scales, a phenomenon best described by fractional-order temporal dynamics. This motivates the use of fractional derivatives for modeling neural responses and membrane adaptation. A new fast computational method for evaluating Caputo fractional derivatives is proposed for use when the argument’s interval spans many orders of time. In this approach, the weakly singular kernel of the fractional derivative is approximated by a hierarchical, multiscale weighted sum of exponential functions, thereby converting the original convolution into a system of ordinary differential equations. The formulation ensures numerical stability and allows an efficient parallel implementation, making it suitable for analyzing long and oscillatory signals. Using intracellular data from neocortical pyramidal neurons, we show that the method accurately reproduces firing-rate adaptation and reveals a spectrum of characteristic time scales consistent with ion-channel kinetics and slow electrogenic processes. The proposed approach offers a computationally efficient framework for modeling systems with long-term memory and for processing experimental neurophysiological data.
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© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
