Extraction of parameters of a stochastic integrate-and-fire model with adaptation from voltage recordings

Biological Cybernetics (2025) 119:2 https://doi.org/10.1007/s00422-024-01000-2 ORIGINAL ARTICLE Extraction of parameters of a stochastic integrate-and-fire model with adaptation from voltage recordings Lilli Kiessling1,2 · Benjamin Lindner1,3 Received: 7 May 2024 / Accepted: 21 November 2024 © The Author(s) 2024 Abstract Integrate-and-fire models are an important class of phenomenological neuronal models that are frequently used in computational studies of single neural activity, population activity, and recurrent neural networks. If these models are used to understand and interpret electrophysiological data, it is important to reliably estimate the values of the model’s parameters. However, there are no standard methods for the parameter estimation of Integrate-and-fire models. Here, we identify the model parameters of an adaptive integrate-and-fire neuron with temporally correlated noise by analyzing membrane potential and spike trains in response to a current step. Explicit formulas for the parameters are analytically derived by stationary and time-dependent ensemble averaging of the model dynamics. Specifically, we give mathematical expressions for the adaptation time constant, the adaptation strength, the membrane time constant, and the mean constant input current. These theoretical predictions are validated by numerical simulations for a broad range of system parameters. Importantly, we demonstrate that parameters can be extracted by using only a modest number of trials. This is particularly encouraging, as the number of trials in experimental settings is often limited. Hence, our formulas may be useful for the extraction of effective parameters from neurophysiological data obtained from standard current-step experiments. Keywords Stochastic spiking · Integrate-and-fire model · Spike-frequency adaptation · Parameter extraction for neural models 1 Introduction Integrate-and-fire (IF) neuron models are widely used in theoretical studies of neural dynamics (see e.g. Johannesma 1968; Knight 1972; Treves 1993; Campbell et al. 1999; Brunel 2000; Brunel et al. 2001; Lindner et al. 2005; de la Rocha et al. 2007; Litwin-Kumar and Doiron 2012; Lindner 2022) and reviews (Holden 1976; Ricciardi 1977; Tuckwell 1989; Burkitt 2006a, b). These models simplify the complex Communicated by Paul Tiesinga. B Lilli Kiessling Benjamin Lindner 1 Bernstein Center for Computational Neuroscience Berlin, Philippstr. 13, Haus 2, 10115 Berlin, Germany 2 Physics Department of Technische, Universit Berlin, Hardenbergstr. 36, 10623 Berlin, Germany 3 Physics Department, Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany properties of neurons into a manageable framework, making it possible to analyze spontaneous neural activity and predict neural responses to time-dependent stimuli. Although basic in nature, IF models capture the timing of neuronal spikes effectively, which is crucial for understanding how neurons communicate and process information (Gerstner and Naud 2009). The leaky integrate-and-fire (LIF) model (Lapicque 1907; Stein 1967; Tuckwell 1988) combines input integration with a fire-and-reset rule. It was previously shown, that including mechanisms for adaptation is important to capture neural spike process properly (Benda and Herz 2003; Brette and Gerstner 2005). Another important addition is the incorporation of a noise source to account for the notorious stochasticity of spike generation in many situations. Often, the noise that may stem from channel fluctuations or from synaptic inputs is low-pass filtered in time-due to slow channel kinetics (Schwalger et al. 2010; Fisch et al. 2012) and synaptic dynamics (Brunel and Sergi 1998; MorenoBote and Parga 2010), respectively. A standard choice of a model with Gaussian low-pass filtered noise is the stochastic 0123456789().: V,-vol 123 2 Page 2 of 10 Ornstein-Uhlenbeck process (originally introduced to model the velocity of a Brownian particle (Uhlenbeck and Ornstein 1930). Gaussian statistics arise in many situations when an abundance of nearly independent inputs add up - these can be currents through many ion channels or the inputs at many synapses. We mention in passing that other relevant noise statistics in neurons are shot noise (when the spike character of synaptic input cannot be neglected, see e.g. Richardson and Swarbrick 2010; Droste and Lindner 2017; Richardson 2024) or dichotomous noise (when up/down states from a surrounding network dominate the fluctuation input, see e.g. Droste and Lindner 2014; Mankin and Lumi 2016). Accurately identifying model parameters that reflect experimental data is essential for the utility of these models in experimental and theoretical studies (Paninski et al. 2003; Huys et al. 2006; Rossant et al. 2011; Iolov et al. 2017; Ladenbauer et al. 2019; Friedrich et al. 2014). Traditional methods for parameter estimation in IF models often rely on numerical fitting (Friedrich et al. 2014; Teeter et al. 2018). In some experiments in vitro, a noisy current (in the form of a computer-generated Ornstein-Uhlenbeck process) is injected into the cell, which allows to extract subthreshold nonlinearities and their parameters directly; see e.g. the pioneering studies by Badel et al. (2008a, b). Other studies (Vilela and Lindner 2009a, b) have provided relations of the firing statistics of simple IF models with white Gaussian noise, specifically their firing rate and coefficient of variation of the interspike interval (ISI) to the input parameters (base current and noise intensity). Because the neural spiking process is inherently nonlinear, and not all relevant variables are also observable (adaptation currents are difficult to access), the estimation of parameters of spiking neuron models based on experimental data remains a difficult task. In our study, we introduce a new analytical method that derives essential parameters of the adaptive leaky integrateand-fire model with an (unknown) low-pass filtered Gaussian noise. We assume that we know the response of the membrane voltage to a current-step for a sufficiently large number of trials. The method provides the adaptation time constant, adaptation strength, membrane time constant, and mean input current. Importantly, it does not require explicit knowledge of the time course or characteristics of the intrinsic noise, making it applicable to a wide range of experimental conditions. This approach can potentially facilitate the classification of neuron types (Teeter et al. 2018) and the exploration of fluctuation-response relationships in experimental settings (Lindner 2022, 2002b; Puttkammer and Lindner 2024). This paper is structured as follows: we begin by describing the adaptive integrate-and-fire model with OrnsteinUhlenbeck noise, explain the new method for extracting parameters, validate this method with numerical simulations, and, finally, briefly summarize our finding and give an outlook to possible extensions of the (...truncated)