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

Neuronal calcium signals propagating by simple diffusion and reaction with mobile and stationary buffers are limited to cellular microdomains. The distance intracellular calcium signals can travel may be significantly increased by means of calcium-induced calcium release from internal calcium stores, notably the endoplasmic reticulum. The organelle, which can be thought of as a cell-within-a-cell, is able to sequester large amounts of cytosolic calcium ions via SERCA pumps and selectively release them into the cytosol through ryanodine receptor channels leading to the formation of calcium waves. In this study, we set out to investigate the basic properties of such dendritic calcium waves and how they depend on the three parameters dendrite radius, ER radius and ryanodine receptor density in the endoplasmic membrane. We demonstrate that there are stable and abortive regimes for calcium waves, depending on the above morphological and physiological parameters. In stable regimes, calcium waves can travel across long dendritic distances, similar to electrical action potentials. We further observe that abortive regimes exist, which could be relevant for spike-timing dependent plasticity, as travel distances and wave velocities vary with changing intracellular architecture. For some of these regimes, analytic functions could be derived that fit the simulation data. In parameter spaces, that are non-trivially influenced by the three-dimensional calcium concentration profile, we were not able to derive such a functional description, demonstrating the mathematical requirement to model and simulate biochemical signaling in three-dimensional space.

Markus Breit

Gillian Queisser

What Is Required for Neuronal Calcium Waves? A Numerical Parameter Study

神经元钙波需要什么？数值参数研究

We present the study of a minimal microcircuit controlling locomotion in two-day-old Xenopus tadpoles. During swimming, neurons in the spinal central pattern generator (CPG) generate anti-phase oscillations between left and right half-centres. Experimental recordings show that the same CPG neurons can also generate transient bouts of long-lasting in-phase oscillations between left-right centres. These synchronous episodes are rarely recorded and have no identified behavioural purpose. However, metamorphosing tadpoles require both anti-phase and in-phase oscillations for swimming locomotion. Previous models have shown the ability to generate biologically realistic patterns of synchrony and swimming oscillations in tadpoles, but a mathematical description of how these oscillations appear is still missing. We define a simplified model that incorporates the key operating principles of tadpole locomotion. The model generates the various outputs seen in experimental recordings, including swimming and synchrony. To study the model, we perform detailed one- and two-parameter bifurcation analysis. This reveals the critical boundaries that separate different dynamical regimes and demonstrates the existence of parameter regions of bi-stable swimming and synchrony. We show that swimming is stable in a significantly larger range of parameters, and can be initiated more robustly, than synchrony. Our results can explain the appearance of long-lasting synchrony bouts seen in experiments at the start of a swimming episode.

Robert Merrison-Hort

Stephen R. Soffe

Wen-Chang Li

Roman Borisyuk

Bifurcations of Limit Cycles in a Reduced Model of the Xenopus Tadpole Central Pattern Generator

非洲爪蟾蝌蚪中枢模式发生器简化模型中的极限环分岔

This tutorial illustrates the use of data assimilation algorithms to estimate unobserved variables and unknown parameters of conductance-based neuronal models. Modern data assimilation (DA) techniques are widely used in climate science and weather prediction, but have only recently begun to be applied in neuroscience. The two main classes of DA techniques are sequential methods and variational methods. We provide computer code implementing basic versions of a method from each class, the Unscented Kalman Filter and 4D-Var, and demonstrate how to use these algorithms to infer several parameters of the Morris–Lecar model from a single voltage trace. Depending on parameters, the Morris–Lecar model exhibits qualitatively different types of neuronal excitability due to changes in the underlying bifurcation structure. We show that when presented with voltage traces from each of the various excitability regimes, the DA methods can identify parameter sets that produce the correct bifurcation structure even with initial parameter guesses that correspond to a different excitability regime. This demonstrates the ability of DA techniques to perform nonlinear state and parameter estimation and introduces the geometric structure of inferred models as a novel qualitative measure of estimation success. We conclude by discussing extensions of these DA algorithms that have appeared in the neuroscience literature.

Matthew J. Moye

Casey O. Diekman

Data Assimilation Methods for Neuronal State and Parameter Estimation

神经元状态和参数估计的数据同化方法

We review recent work on the theory and applications of stochastic hybrid systems in cellular neuroscience. A stochastic hybrid system or piecewise deterministic Markov process involves the coupling between a piecewise deterministic differential equation and a time-homogeneous Markov chain on some discrete space. The latter typically represents some random switching process. We begin by summarizing the basic theory of stochastic hybrid systems, including various approximation schemes in the fast switching (weak noise) limit. In subsequent sections, we consider various applications of stochastic hybrid systems, including stochastic ion channels and membrane voltage fluctuations, stochastic gap junctions and diffusion in randomly switching environments, and intracellular transport in axons and dendrites. Finally, we describe recent work on phase reduction methods for stochastic hybrid limit cycle oscillators.

Paul C. Bressloff

James N. Maclaurin

Stochastic Hybrid Systems in Cellular Neuroscience

细胞神经科学中的随机混合系统

Theta (4–8 Hz) and gamma (30–80 Hz) rhythms in the brain are commonly associated with memory and learning (Kahana in J Neurosci 26:1669–1672, 2006; Quilichini et al. in J Neurosci 30:11128–11142, 2010). The precision of co-firing between neurons and incoming inputs is critical in these cognitive functions. We consider an inhibitory neuron model with M-current under forcing from gamma pulses and a sinusoidal current of theta frequency. The M-current has a long time constant (∼90 ms) and it has been shown to generate resonance at theta frequencies (Hutcheon and Yarom in Trends Neurosci 23:216–222, 2000; Hu et al. in J Physiol 545:783–805, 2002). We have found that this slow M-current contributes to the precise co-firing between the network and fast gamma pulses in the presence of a slow sinusoidal forcing. The M-current expands the phase-locking frequency range of the network, counteracts the slow theta forcing, and admits bistability in some parameter range. The effects of the M-current balancing the theta forcing are reduced if the sinusoidal current is faster than the theta frequency band. We characterize the dynamical mechanisms underlying the role of the M-current in enabling a network to be entrained to gamma frequency inputs using averaging methods, geometric singular perturbation theory, and bifurcation analysis.

Yujia Zhou

Theodore Vo

Horacio G. Rotstein

Michelle M. McCarthy

Nancy Kopell

M-Current Expands the Range of Gamma Frequency Inputs to Which a Neuronal Target Entrains

M电流扩展了神经目标同步的伽马频率输入范围

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)