Fluctuation-response relations for a two-stage population of spiking neurons stimulated by common noise - Leander Dittrich; Benjamin Lindner (Biological Cybernetics, 2026)
Recently a method has been put forward to connect the measures of spontaneous neuronal activity and the measures of the average single-neuron response to stimuli via fluctuation-response relations (FRRs) for some integrate-and-fire (IF) type neuron models. In this work we expand this method to populations of neurons, relating their spontaneous correlation and linear-response statistics. To this end, we analyze the simple case of uncoupled cells modeled by IF neurons (first stage of processing) which receive common stochastic input and project their output spike trains onto a readout neuron (second stage of processing). We derive and verify FRRs connecting the single neuron response to cross-correlations among neurons and the response of the full system to cross-stage correlations. Furthermore, we utilize these FRRs to derive approximations of all cross-stage cross-spectra for a relevant model of a second-stage cell, the partial synchronous output (PSO). We conclude with a discussion of how our results can be expanded to more involved network settings and neuron models.

Effect of burst spikes on linear and nonlinear signal transmission in spiking neurons - Maria Schlungbaum; Alexandra Barayeu; Jan Grewe; Jan Benda; Benjamin Lindner (Journal of Computational Neuroscience, 2025)
We study the impact of bursts on spike statistics and neural signal transmission. We propose a stochastic burst algorithm that is applied to a burst-free spike train and adds a random number of temporally-jittered burst spikes to each spike. This simple algorithm ignores any possible stimulus-dependence of bursting but allows to relate spectra and signal-transmission characteristics of burst-free and burst-endowed spike trains. By averaging over the various statistical ensembles, we find a frequency-dependent factor connecting the linear and also the second-order susceptibility of the spike trains with and without bursts. The relation between spectra is more complicated: besides a frequency-dependent multiplicative factor it also involves an additional frequency-dependent offset. We confirm these relations for the (burst-free) spike trains of a stochastic integrate-and-fire neuron and identify frequency ranges in which the transmission is boosted or diminished by bursting. We then consider bursty spike trains of electroreceptor afferents of weakly electric fish and approach the role of burst spikes as follows. We compare the spectral statistics of the bursty spike train to (i) that of a spike train with burst spikes removed and to (ii) that of the spike train in (i) endowed by bursts according to our algorithm. Significant spectral features are explained by our signal-independent burst algorithm, e.g. the burst-induced boosting of the nonlinear response. A difference is seen in the information transfer for the original bursty spike train and our burst-endowed spike train. Our algorithm is thus helpful to identify different effects of bursting.

Extraction of parameters of a stochastic integrate-and-fire model with adaptation from voltage recordings - Lilli Kiessling; Benjamin Lindner (Biological Cybernetics, 2024)
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.

Beyond the Nobel prizes: towards new synergies between Computational Neuroscience and Artificial Intelligence - Jean-Marc Fellous; Peter Thomas; Paul Tiesinga; Benjamin Lindner (Biological Cybernetics, 2024)

Fluctuation–response relations for integrate-and-fire models with an absolute refractory period - Friedrich Puttkammer; Benjamin Lindner (Biological Cybernetics, 2024)
We study the problem of relating the spontaneous fluctuations of a stochastic integrate-and-fire (IF) model to the response of the instantaneous firing rate to time-dependent stimulation if the IF model is endowed with a non-vanishing refractory period and a finite (stereotypical) spike shape. This seemingly harmless addition to the model is shown to complicate the analysis put forward by Lindner Phys. Rev. Lett. (2022), i.e., the incorporation of the reset into the model equation, the Rice-like averaging of the stochastic differential equation, and the application of the Furutsu–Novikov theorem. We derive a still exact (although more complicated) fluctuation–response relation (FRR) for an IF model with refractory state and a white Gaussian background noise. We also briefly discuss an approximation for the case of a colored Gaussian noise and conclude with a summary and outlook on open problems.

The steady state and response to a periodic stimulation of the firing rate for a theta neuron with correlated noise - Jannik Franzen; Lukas Ramlow; Benjamin Lindner (Journal of Computational Neuroscience, 2023)
The stochastic activity of neurons is caused by various sources of correlated fluctuations and can be described in terms of simplified, yet biophysically grounded, integrate-and-fire models. One paradigmatic model is the quadratic integrate-and-fire model and its equivalent phase description by the theta neuron. Here we study the theta neuron model driven by a correlated Ornstein-Uhlenbeck noise and by periodic stimuli. We apply the matrix-continued-fraction method to the associated Fokker-Planck equation to develop an efficient numerical scheme to determine the stationary firing rate as well as the stimulus-induced modulation of the instantaneous firing rate. For the stationary case, we identify the conditions under which the firing rate decreases or increases by the effect of the colored noise and compare our results to existing analytical approximations for limit cases. For an additional periodic signal we demonstrate how the linear and nonlinear response terms can be computed and report resonant behavior for some of them. We extend the method to the case of two periodic signals, generally with incommensurable frequencies, and present a particular case for which a strong mixed response to both signals is observed, i.e. where the response to the sum of signals differs significantly from the sum of responses to the single signals. We provide Python code for our computational method: https://github.com/jannikfranzen/theta_neuron .

Quantitative comparison of the mean–return-time phase and the stochastic asymptotic phase for noisy oscillators - Alberto Pérez-Cervera; Benjamin Lindner; Peter J. Thomas (Biological Cybernetics, 2022)
Seminal work by A. Winfree and J. Guckenheimer showed that a deterministic phase variable can be defined either in terms of Poincaré sections or in terms of the asymptotic (long-time) behaviour of trajectories approaching a stable limit cycle. However, this equivalence between the deterministic notions of phase is broken in the presence of noise. Different notions of phase reduction for a stochastic oscillator can be defined either in terms of mean–return-time sections or as the argument of the slowest decaying complex eigenfunction of the Kolmogorov backwards operator. Although both notions of phase enjoy a solid theoretical foundation, their relationship remains unexplored. Here, we quantitatively compare both notions of stochastic phase. We derive an expression relating both notions of phase and use it to discuss differences (and similarities) between both definitions of stochastic phase for (i) a spiral sink motivated by stochastic models for electroencephalograms, (ii) noisy limit-cycle systems-neuroscience models, and (iii) a stochastic heteroclinic oscillator inspired by a simple motor-control system.

Mean-return-time phase of a stochastic oscillator provides an approximate renewal description for the associated point process - Konstantin Holzhausen; Lukas Ramlow; Shusen Pu; Peter J. Thomas; Benjamin Lindner (Biological Cybernetics, 2022)
Stochastic oscillations can be characterized by a corresponding point process; this is a common practice in computational neuroscience, where oscillations of the membrane voltage under the influence of noise are often analyzed in terms of the interspike interval statistics, specifically the distribution and correlation of intervals between subsequent threshold-crossing times. More generally, crossing times and the corresponding interval sequences can be introduced for different kinds of stochastic oscillators that have been used to model variability of rhythmic activity in biological systems. In this paper we show that if we use the so-called mean-return-time (MRT) phase isochrons (introduced by Schwabedal and Pikovsky) to count the cycles of a stochastic oscillator with Markovian dynamics, the interphase interval sequence does not show any linear correlations, i.e., the corresponding sequence of passage times forms approximately a renewal point process. We first outline the general mathematical argument for this finding and illustrate it numerically for three models of increasing complexity: (i) the isotropic Guckenheimer–Schwabedal–Pikovsky oscillator that displays positive interspike interval (ISI) correlations if rotations are counted by passing the spoke of a wheel; (ii) the adaptive leaky integrate-and-fire model with white Gaussian noise that shows negative interspike interval correlations when spikes are counted in the usual way by the passage of a voltage threshold; (iii) a Hodgkin–Huxley model with channel noise (in the diffusion approximation represented by Gaussian noise) that exhibits weak but statistically significant interspike interval correlations, again for spikes counted when passing a voltage threshold. For all these models, linear correlations between intervals vanish when we count rotations by the passage of an MRT isochron. We finally discuss that the removal of interval correlations does not change the long-term variability and its effect on information transmission, especially in the neural context.

Induced seismicity due to hydraulic fracturing near Blackpool, UK: source modeling and event detection - Nasim Karamzadeh; Mike Lindner; Benjamin Edwards; Emmanuel Gaucher; Andreas Rietbrock (Journal of Seismology, 2021)
Monitoring small magnitude induced seismicity requires a dense network of seismic stations and high-quality recordings in order to precisely determine events’ hypocentral parameters and mechanisms. However, microseismicity (e.g. swarm activity) can also occur in an area where a dense network is unavailable and recordings are limited to a few seismic stations at the surface. In this case, using advanced event detection techniques such as template matching can help to detect small magnitude shallow seismic events and give insights about the ongoing process at the subsurface giving rise to microseismicity. In this paper, we study shallow microseismic events caused by hydrofracking of the PNR-2 well near Blackpool, UK, in 2019 using recordings of a seismic network which was not designed to detect and locate such small events. By utilizing a sparse network of surface stations, small seismic events are detected using template matching technique. In addition, we apply a full-waveform moment tensor inversion to study the focal mechanisms of larger events ( M L > 1) and used the double-difference location technique for events with high-quality and similar waveforms to obtain accurate relative locations. During the stimulation period, temporal changes in event detection rate were in agreement with injection times. Focal mechanisms of the events with high-quality recordings at multiple stations indicate a strike-slip mechanism, while a cross-section of 34 relocated events matches the dip angle of the active fault.

Recurrence-mediated suprathreshold stochastic resonance - Gregory Knoll; Benjamin Lindner (Journal of Computational Neuroscience, 2021)
It has previously been shown that the encoding of time-dependent signals by feedforward networks (FFNs) of processing units exhibits suprathreshold stochastic resonance (SSR), which is an optimal signal transmission for a finite level of independent, individual stochasticity in the single units. In this study, a recurrent spiking network is simulated to demonstrate that SSR can be also caused by network noise in place of intrinsic noise. The level of autonomously generated fluctuations in the network can be controlled by the strength of synapses, and hence the coding fraction (our measure of information transmission) exhibits a maximum as a function of the synaptic coupling strength. The presence of a coding peak at an optimal coupling strength is robust over a wide range of individual, network, and signal parameters, although the optimal strength and peak magnitude depend on the parameter being varied. We also perform control experiments with an FFN illustrating that the optimized coding fraction is due to the change in noise level and not from other effects entailed when changing the coupling strength. These results also indicate that the non-white (temporally correlated) network noise in general provides an extra boost to encoding performance compared to the FFN driven by intrinsic white noise fluctuations.

Biological Cybernetics: 60 years and more to come - Benjamin Lindner; Peter J. Thomas; Jean-Marc Fellous; Paul Tiesinga (Biological Cybernetics, 2021)

A Renewed Vision for Biological Cybernetics - Lindner Benjamin; Peter J. Thomas; Jean-Marc Fellous (Biological Cybernetics, 2020)

Information filtering by coincidence detection of synchronous population output: analytical approaches to the coherence function of a two-stage neural system - Žiga Bostner; Gregory Knoll; Benjamin Lindner (Biological Cybernetics, 2020)
Information about time-dependent sensory stimuli is encoded in the activity of neural populations; distinct aspects of the stimulus are read out by different types of neurons: while overall information is perceived by integrator cells, so-called coincidence detector cells are driven mainly by the synchronous activity in the population that encodes predominantly high-frequency content of the input signal (high-pass information filtering). Previously, an analytically accessible statistic called the partial synchronous output was introduced as a proxy for the coincidence detector cell’s output in order to approximate its information transmission. In the first part of the current paper, we compare the information filtering properties (specifically, the coherence function) of this proxy to those of a simple coincidence detector neuron. We show that the latter’s coherence function can indeed be well-approximated by the partial synchronous output with a time scale and threshold criterion that are related approximately linearly to the membrane time constant and firing threshold of the coincidence detector cell. In the second part of the paper, we propose an alternative theory for the spectral measures (including the coherence) of the coincidence detector cell that combines linear-response theory for shot-noise driven integrate-and-fire neurons with a novel perturbation ansatz for the spectra of spike-trains driven by colored noise. We demonstrate how the variability of the synaptic weights for connections from the population to the coincidence detector can shape the information transmission of the entire two-stage system.

Improved lower bound for the mutual information between signal and neural spike count - Sergej O. Voronenko; Benjamin Lindner (Biological Cybernetics, 2018)

Welcome from the new Editor(s)-in-Chief - Benjamin Lindner; Peter J. Thomas; Jean-Marc Fellous (Biological Cybernetics, 2018)

Coding of time-dependent stimuli in homogeneous and heterogeneous neural populations - Manuel Beiran; Alexandra Kruscha; Jan Benda; Benjamin Lindner (Journal of Computational Neuroscience, 2018)

A cross-scale impact assessment of European nature protection policies under contrasting future socio-economic pathways - Hermann Lotze-Campen; Peter H. Verburg; Alexander Popp; Marcus Lindner; Pieter J. Verkerk; Alexander Moiseyev; Elizabeth Schrammeijer; John Helming; Andrzej Tabeau; Catharina J. E. Schulp; Emma H. van der Zanden; Carlo Lavalle; et al. (Regional Environmental Change, 2018)

Exact analytical results for integrate-and-fire neurons driven by excitatory shot noise - Felix Droste; Benjamin Lindner (Journal of Computational Neuroscience, 2017)

Information filtering in resonant neurons - Sven Blankenburg; Wei Wu; Benjamin Lindner; Susanne Schreiber (Journal of Computational Neuroscience, 2015)

Statistical structure of neural spiking under non-Poissonian or other non-white stimulation - Tilo Schwalger; Felix Droste; Benjamin Lindner (Journal of Computational Neuroscience, 2015)

Erratum to: Information filtering by synchronous spikes in a neural population - Nahal Sharafi; Jan Benda; Benjamin Lindner (Journal of Computational Neuroscience, 2015)

Interspike interval correlation in a stochastic exponential integrate-and-fire model with subthreshold and spike-triggered adaptation - LieJune Shiau; Tilo Schwalger; Benjamin Lindner (Journal of Computational Neuroscience, 2015)

Integrate-and-fire neurons driven by asymmetric dichotomous noise - Felix Droste; Benjamin Lindner (Biological Cybernetics, 2014)

Information filtering by synchronous spikes in a neural population - Nahal Sharafi; Jan Benda; Benjamin Lindner (Journal of Computational Neuroscience, 2013)

Threshold fatigue and information transfer - Maurice J. Chacron; Benjamin Lindner; André Longtin (Journal of Computational Neuroscience, 2007)