Abstract Mathematical models at multiple temporal and spatial scales can unveil the fundamental mechanisms of critical transitions in brain activities. Neural mass models (NMMs) consider the average temporal dynamics of interconnected neuronal subpopulations without explicitly representing the underlying cellular activity. The mesoscopic level offered by the neural mass formulation has been used to model electroencephalographic (EEG) recordings and to investigate various cerebral mechanisms, such as the generation of physiological and pathological brain activities. In this work, we consider a NMM widely accepted in the context of epilepsy, which includes four interacting neuronal subpopulations with different synaptic kinetics. Due to the resulting three-time-scale structure, the model yields complex oscillations of relaxation and bursting types. By applying the principles of geometric singular perturbation theory, we unveil the existence of the canard solutions and detail how they organize the complex oscillations and excitability properties of the model. In particular, we show that boundaries between pathological epileptic discharges and physiological background activity are determined by the canard solutions. Finally we report the existence of canard-mediated small-amplitude frequency-specific oscillations in simulated local field potentials for decreased inhibition conditions. Interestingly, such oscillations are actually observed in intracerebral EEG signals recorded in epileptic patients during pre-ictal periods, close to seizure onsets.

Elif Köksal Ersöz

Fabrice Wendling

The Journal of Mathematical Neuroscience

Canard solutions in neural mass models: consequences on critical regimes

神经质量模型中的卡纳德解：对临界状态的影响

Abstract The principle of stationary action is a cornerstone of modern physics, providing a powerful framework for investigating dynamical systems found in classical mechanics through to quantum field theory. However, computational neuroscience, despite its heavy reliance on concepts in physics, is anomalous in this regard as its main equations of motion are not compatible with a Lagrangian formulation and hence with the principle of stationary action. Taking the Dynamic Causal Modelling (DCM) neuronal state equation as an instructive archetype of the first-order linear differential equations commonly found in computational neuroscience, we show that it is possible to make certain modifications to this equation to render it compatible with the principle of stationary action. Specifically, we show that a Lagrangian formulation of the DCM neuronal state equation is facilitated using a complex dependent variable, an oscillatory solution, and a Hermitian intrinsic connectivity matrix. We first demonstrate proof of principle by using Bayesian model inversion to show that both the original and modified models can be correctly identified via in silico data generated directly from their respective equations of motion. We then provide motivation for adopting the modified models in neuroscience by using three different types of publicly available in vivo neuroimaging datasets, together with open source MATLAB code, to show that the modified (oscillatory) model provides a more parsimonious explanation for some of these empirical timeseries. It is our hope that this work will, in combination with existing techniques, allow people to explore the symmetries and associated conservation laws within neural systems – and to exploit the computational expediency facilitated by direct variational techniques.

Erik D. Fagerholm

W. M. C. Foulkes

Karl J. Friston

Rosalyn J. Moran

Robert Leech

Rendering neuronal state equations compatible with the principle of stationary action

使神经元状态方程与最小作用量原理相容

Abstract We study pattern formation in a 2-population homogenized neural field model of the Hopfield type in one spatial dimension with periodic microstructure. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. It is shown that the nonlocal synaptic interactions promote a finite band width instability. The stability method relies on a sequence of wave-number dependent invariants of $2\times 2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn><mml:mo>×</mml:mo><mml:mn>2</mml:mn></mml:math> -stability matrices representing the sequence of Fourier-transformed linearized evolution equations for the perturbation imposed on the homogeneous background. The generic picture of the instability structure consists of a finite set of well-separated gain bands. In the shallow firing rate regime the nonlinear development of the instability is determined by means of the translational invariant model with connectivity kernels replaced with the corresponding period averaged connectivity functions. In the steep firing rate regime the pattern formation process depends sensitively on the spatial localization of the connectivity kernels: For strongly localized kernels this process is determined by the translational invariant model with period averaged connectivity kernels, whereas in the complementary regime of weak and moderate localization requires the homogenized model as a starting point for the analysis. We follow the development of the instability numerically into the nonlinear regime for both steep and shallow firing rate functions when the connectivity kernels are modeled by means of an exponentially decaying function. We also study the pattern forming process numerically as a function of the heterogeneity parameters in four different regimes ranging from the weakly modulated case to the strongly heterogeneous case. For the weakly modulated regime, we observe that stable spatial oscillations are formed in the steep firing rate regime, whereas we get spatiotemporal oscillations in the shallow regime of the firing rate functions.

Karina Kolodina

John Wyller

Anna Oleynik

Mads Peter Sørensen

Pattern formation in a 2-population homogenized neuronal network model

两群体均质化神经网络模型中的模式形成

Abstract In the auditory streaming paradigm, alternating sequences of pure tones can be perceived as a single galloping rhythm (integration) or as two sequences with separated low and high tones (segregation). Although studied for decades, the neural mechanisms underlining this perceptual grouping of sound remains a mystery. With the aim of identifying a plausible minimal neural circuit that captures this phenomenon, we propose a firing rate model with two periodically forced neural populations coupled by fast direct excitation and slow delayed inhibition. By analyzing the model in a non-smooth, slow-fast regime we analytically prove the existence of a rich repertoire of dynamical states and of their parameter dependent transitions. We impose plausible parameter restrictions and link all states with perceptual interpretations. Regions of stimulus parameters occupied by states linked with each percept match those found in behavioural experiments. Our model suggests that slow inhibition masks the perception of subsequent tones during segregation (forward masking), whereas fast excitation enables integration for large pitch differences between the two tones.

Andrea Ferrario

James Rankin

Auditory streaming emerges from fast excitation and slow delayed inhibition

听觉流从快速兴奋和缓慢延迟抑制中产生

Abstract The neurons of the deep cerebellar nuclei (DCNn) represent the main functional link between the cerebellar cortex and the rest of the central nervous system. Therefore, understanding the electrophysiological properties of DCNn is of fundamental importance to understand the overall functioning of the cerebellum. Experimental data suggest that DCNn can reversibly switch between two states: the firing of spikes (F state) and a stable depolarized state (SD state). We introduce a new biophysical model of the DCNn membrane electro-responsiveness to investigate how the interplay between the documented conductances identified in DCNn give rise to these states. In the model, the F state emerges as an isola of limit cycles, i.e. a closed loop of periodic solutions disconnected from the branch of SD fixed points. This bifurcation structure endows the model with the ability to reproduce the $\text{F}\to \text{SD}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>F</mml:mtext> <mml:mo>→</mml:mo> <mml:mtext>SD</mml:mtext> </mml:math> transition triggered by hyperpolarizing current pulses. The model also reproduces the $\text{F}\to \text{SD}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>F</mml:mtext> <mml:mo>→</mml:mo> <mml:mtext>SD</mml:mtext> </mml:math> transition induced by blocking Ca currents and ascribes this transition to the blocking of the high-threshold Ca current. The model suggests that intracellular current injections can trigger fully reversible $\text{F}\leftrightarrow \text{SD}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>F</mml:mtext> <mml:mo>↔</mml:mo> <mml:mtext>SD</mml:mtext> </mml:math> transitions. Investigation of low-dimension reduced models suggests that the voltage-dependent Na current is prominent for these dynamical features. Finally, simulations of the model suggest that physiological synaptic inputs may trigger $\text{F}\leftrightarrow \text{SD}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>F</mml:mtext> <mml:mo>↔</mml:mo> <mml:mtext>SD</mml:mtext> </mml:math> transitions. These transitions could explain the puzzling observation of positively correlated activities of connected Purkinje cells and DCNn despite the former inhibit the latter.

Hugues Berry

Stéphane Genet

A model of on/off transitions in neurons of the deep cerebellar nuclei: deciphering the underlying ionic mechanisms

深部小脑核神经元开/关转换模型：揭示潜在的离子机制

Abstract Decoding approaches provide a useful means of estimating the information contained in neuronal circuits. In this work, we analyze the expected classification error of a decoder based on Fisher linear discriminant analysis. We provide expressions that relate decoding error to the specific parameters of a population model that performs linear integration of sensory input. Results show conditions that lead to beneficial and detrimental effects of noise correlation on decoding. Further, the proposed framework sheds light on the contribution of neuronal noise, highlighting cases where, counter-intuitively, increased noise may lead to improved decoding performance. Finally, we examined the impact of dynamical parameters, including neuronal leak and integration time constant, on decoding. Overall, this work presents a fruitful approach to the study of decoding using a comprehensive theoretical framework that merges dynamical parameters with estimates of readout error.

Matias Calderini

Jean-Philippe Thivierge

Estimating Fisher discriminant error in a linear integrator model of neural population activity

估计神经群体活动线性积分器模型中的Fisher判别误差

Abstract In this work, we consider a general conductance-based neuron model with the inclusion of the acetycholine sensitive, M-current. We study bifurcations in the parameter space consisting of the applied current $I_{app}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>p</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> </mml:math> , the maximal conductance of the M-current $g_{M}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:math> and the conductance of the leak current $g_{L}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:math> . We give precise conditions for the model that ensure the existence of a Bogdanov–Takens (BT) point and show that such a point can occur by varying $I_{app}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>p</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> </mml:math> and $g_{M}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:math> . We discuss the case when the BT point becomes a Bogdanov–Takens–cusp (BTC) point and show that such a point can occur in the three-dimensional parameter space. The results of the bifurcation analysis are applied to different neuronal models and are verified and supplemented by numerical bifurcation diagrams generated using the package . We conclude that there is a transition in the neuronal excitability type organised by the BT point and the neuron switches from Class-I to Class-II as conductance of the M-current increases.

Isam Al-Darabsah

Sue Ann Campbell

M-current induced Bogdanov–Takens bifurcation and switching of neuron excitability class

M电流诱导的Bogdanov-Takens分岔与神经元兴奋性类别的切换

Memory and forgetting constitute two sides of the same coin, and although the first has been extensively investigated, the latter is often overlooked. A possible approach to better understand forgetting is to develop phenomenological models that implement its putative mechanisms in the most elementary way possible, and then experimentally test the theoretical predictions of these models. One such mechanism proposed in previous studies is retrograde interference, stating that a memory can be erased due to subsequently acquired memories. In the current contribution, we hypothesize that retrograde erasure is controlled by the relevant “importance” measures such that more important memories eliminate less important ones acquired earlier. We show that some versions of the resulting mathematical model are broadly compatible with the previously reported power-law forgetting time course and match well the results of our recognition experiments with long, randomly assembled streams of words.

Antonios Georgiou

Mikhail Katkov

Misha Tsodyks

J. Math. Neurosc.

Retroactive interference model of forgetting

遗忘的倒摄干扰模型

Abstract We analyse the potential effects of lateral connectivity (amacrine cells and gap junctions) on motion anticipation in the retina. Our main result is that lateral connectivity can—under conditions analysed in the paper—trigger a wave of activity enhancing the anticipation mechanism provided by local gain control (Berry et al. in Nature 398(6725):334–338, 1999; Chen et al. in J. Neurosci. 33(1):120–132, 2013). We illustrate these predictions by two examples studied in the experimental literature: differential motion sensitive cells (Baccus and Meister in Neuron 36(5):909–919, 2002) and direction sensitive cells where direction sensitivity is inherited from asymmetry in gap junctions connectivity (Trenholm et al. in Nat. Neurosci. 16:154–156, 2013). We finally present reconstructions of retinal responses to 2D visual inputs to assess the ability of our model to anticipate motion in the case of three different 2D stimuli.

Selma Souihel

Bruno Cessac

On the potential role of lateral connectivity in retinal anticipation

视网膜预期中横向连接的潜在作用

Abstract The brain is intrinsically organized into large-scale networks that constantly re-organize on multiple timescales, even when the brain is at rest. The timing of these dynamics is crucial for sensation, perception, cognition, and ultimately consciousness, but the underlying dynamics governing the constant reorganization and switching between networks are not yet well understood. Electroencephalogram (EEG) microstates are brief periods of stable scalp topography that have been identified as the electrophysiological correlate of functional magnetic resonance imaging defined resting-state networks. Spatiotemporal microstate sequences maintain high temporal resolution and have been shown to be scale-free with long-range temporal correlations. Previous attempts to model EEG microstate sequences have failed to capture this crucial property and so cannot fully capture the dynamics; this paper answers the call for more sophisticated modeling approaches. We present a dynamical model that exhibits a noisy network attractor between nodes that represent the microstates. Using an excitable network between four nodes, we can reproduce the transition probabilities between microstates but not the heavy tailed residence time distributions. We present two extensions to this model: first, an additional hidden node at each state; second, an additional layer that controls the switching frequency in the original network. Introducing either extension to the network gives the flexibility to capture these heavy tails. We compare the model generated sequences to microstate sequences from EEG data collected from healthy subjects at rest. For the first extension, we show that the hidden nodes ‘trap’ the trajectories allowing the control of residence times at each node. For the second extension, we show that two nodes in the controlling layer are sufficient to model the long residence times. Finally, we show that in addition to capturing the residence time distributions and transition probabilities of the sequences, these two models capture additional properties of the sequences including having interspersed long and short residence times and long range temporal correlations in line with the data as measured by the Hurst exponent.

Jennifer Creaser

Peter Ashwin

Claire Postlethwaite

Juliane Britz

Noisy network attractor models for transitions between EEG microstates

脑电微状态转换的噪声网络吸引子模型

数学神经科学杂志 (JMN) 发表关于神经科学所有领域的数学建模和分析的研究文章，即神经系统及其功能障碍的研究。重点是使用数学作为主要工具来阐明负责在所有相关尺度（从分子世界到认知世界）的神经科学中实验观察到的行为的基本机制。其目的是发表使用先进数学技术来阐明这些问题的作品。它发表完整的原创论文、快速交流和评论文章。特别鼓励结合令人信服的数值实验支持的理论结果的论文。也欢迎介绍和帮助开发可能与未来神经系统特别是人类大脑研究相关的新数学理论的论文.

The Journal of Mathematical Neuroscience (JMN) is dedicated to publishing research articles on mathematical modeling and analysis across all fields of neuroscience, focusing on the study of the nervous system and its disorders. The journal emphasizes the use of mathematics as a primary tool to elucidate the fundamental mechanisms behind experimentally observed behaviors in neuroscience, spanning from the molecular to the cognitive level. We aim to publish original research papers, rapid communications, and review articles that employ advanced mathematical techniques to address these issues. We particularly encourage submissions that present theoretical results supported by compelling numerical experiments. Additionally, we welcome papers that introduce and help develop new mathematical theories that may be relevant to future research on the nervous system, especially the human brain.

《Journal of Mathematical Neuroscience》是一本专注于神经科学各个领域的数学建模与分析研究的期刊。我们致力于使用数学工具揭示神经系统及其功能障碍的基本机制，涵盖从分子层面到认知层面的实验观察。期刊欢迎发表使用先进数学技术的原创研究论文、快速交流和评论文章，特别鼓励那些结合有力数值实验支持的理论成果的研究。同时，我们也欢迎介绍和发展与未来神经系统研究，尤其是人类大脑相关的新数学理论的论文。

Journals	Journal of Mathematical Neuroscience
ISSN	2190-8567
h-index	22
Journal Impact	2.16
Quartiles(Global)	W_MULTIPLE(Q2)
Quartiles(China)	医学(4区)医学数学与计算生物学(3区)医学数学跨学科应用(3区)医学神经科学(4区)