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。

Journal of Mathematical Neuroscience

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

The Journal of Mathematical Neuroscience

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

Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron models typically involve observable differences in time scales. The Hodgkin–Huxley model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5–32, 2007) nondimensionalized this model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio ε. The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type. In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in $\mathbb{R}^{4}$ . We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin–Huxley model in organizing MMOs and determining the firing rates of action potentials. For ε sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the model.

Cris R. Hasan

Bernd Krauskopf

Hinke M. Osinga

Saddle Slow Manifolds and Canard Orbits in R 4 $\mathbb{R}^{4}$ and Application to the Full Hodgkin–Huxley Model

鞍慢流形与Canard轨道在$\mathbb{R}^{4}$中的应用及对完整Hodgkin-Huxley模型的研究

The theory of attractor neural networks has been influential in our understanding of the neural processes underlying spatial, declarative, and episodic memory. Many theoretical studies focus on the inherent properties of an attractor, such as its structure and capacity. Relatively little is known about how an attractor neural network responds to external inputs, which often carry conflicting information about a stimulus. In this paper we analyze the behavior of an attractor neural network driven by two conflicting external inputs. Our focus is on analyzing the emergent properties of the megamap model, a quasi-continuous attractor network in which place cells are flexibly recombined to represent a large spatial environment. In this model, the system shows a sharp transition from the winner-take-all mode, which is characteristic of standard continuous attractor neural networks, to a combinatorial mode in which the equilibrium activity pattern combines embedded attractor states in response to conflicting external inputs. We derive a numerical test for determining the operational mode of the system a priori. We then derive a linear transformation from the full megamap model with thousands of neurons to a reduced 2-unit model that has similar qualitative behavior. Our analysis of the reduced model and explicit expressions relating the parameters of the reduced model to the megamap elucidate the conditions under which the combinatorial mode emerges and the dynamics in each mode given the relative strength of the attractor network and the relative strength of the two conflicting inputs. Although we focus on a particular attractor network model, we describe a set of conditions under which our analysis can be applied to more general attractor neural networks.

Kathryn Hedrick

Kechen Zhang

Analysis of an Attractor Neural Network’s Response to Conflicting External Inputs

吸引子神经网络对冲突外部输入响应的分析

Neurons in a micro-circuit connected by chemical synapses can have their connectivity affected by the prior activity of the cells. The number of synapses available for releasing neurotransmitter can be decreased by repetitive activation through depletion of readily releasable neurotransmitter (NT), or increased through facilitation, where the probability of release of NT is increased by prior activation. These competing effects can create a complicated and subtle range of time-dependent connectivity. Here we investigate the probabilistic properties of facilitation and depression (FD) for a presynaptic neuron that is receiving a Poisson spike train of input. We use a model of FD that is parameterized with experimental data from a hippocampal basket cell and pyramidal cell connection, for fixed frequency input spikes at frequencies in the range of theta (3-8 Hz) and gamma (20-100 Hz) oscillations. Hence our results will apply to micro-circuits in the hippocampus that are responsible for the interaction of theta and gamma rhythms associated with learning and memory. A control situation is compared with one in which a pharmaceutical neuromodulator (muscarine) is employed. We apply standard information-theoretic measures such as entropy and mutual information, and find a closed form approximate expression for the probability distribution of release probability. We also use techniques that measure the dependence of the response on the exact history of stimulation the synapse has received, which uncovers some unexpected differences between control and muscarine-added cases.

Elham Bayat Mokhtari

J. Josh Lawrence

Emily F. Stone

Effect of Neuromodulation of Short-term Plasticity on Information Processing in Hippocampal Interneuron Synapses

短期可塑性神经调控对海马中间神经元突触信息处理的影响

The structure of spiking activity in cortical networks has important implications for how the brain ultimately codes sensory signals. However, our understanding of how network and intrinsic cellular mechanisms affect spiking is still incomplete. In particular, whether cell pairs in a neural network show a positive (or no) relationship between pairwise spike count correlation and average firing rate is generally unknown. This relationship is important because it has been observed experimentally in some sensory systems, and it can enhance information in a common population code. Here we extend our prior work in developing mathematical tools to succinctly characterize the correlation and firing rate relationship in heterogeneous coupled networks. We find that very modest changes in how heterogeneous networks occupy parameter space can dramatically alter the correlation–firing rate relationship.

Andrea K. Barreiro

Cheng Ly

Investigating the Correlation–Firing Rate Relationship in Heterogeneous Recurrent Networks

探究异质性递归网络中的相关性-发放率关系

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

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 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.

Abbreviation	J. Math. Neurosci.
Journal Impact	2.16
Quartiles(Global)	W_MULTIPLE(Q2)

ISSN	2190-8567
h-index	22

Publication Information	Publisher: Springer Verlag，Publishing cycle: ，Journal Type: journal，Open Access Journals: No
Basic data	Year of publication: 2012，Proportion of original research papers: 0.00%，Self Citation Rate:N.A.， Gold OA Rate: 100.00%
Average review cycle	网友分享经验：13 Weeks

Journal of Mathematical Neuroscience

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Auditory streaming emerges from fast excitation and slow delayed inhibition

Saddle Slow Manifolds and Canard Orbits in R 4 $\mathbb{R}^{4}$ and Application to the Full Hodgkin–Huxley Model

Analysis of an Attractor Neural Network’s Response to Conflicting External Inputs

Effect of Neuromodulation of Short-term Plasticity on Information Processing in Hippocampal Interneuron Synapses

Investigating the Correlation–Firing Rate Relationship in Heterogeneous Recurrent Networks

Canard solutions in neural mass models: consequences on critical regimes

Rendering neuronal state equations compatible with the principle of stationary action

Pattern formation in a 2-population homogenized neuronal network model

Auditory streaming emerges from fast excitation and slow delayed inhibition

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