The brain encodes various information through the interaction between macro‐scale phenomena such as Local Field Potential and micro‐scale phenomena such as individual neuron action potential time series. However, the interaction between micro‐ and macro‐scale phenomena that affects episodic memory remains unexplained. We propose a new mathematical framework to predict the parameter conditions under which theta phase precession occurs by constructing models that reproduce the micro‐ and macro‐scale phenomena separately and deriving their phase equations. It was found that phase locking or phase shift between the micro‐scale phase ϕitalicmicro and the macro‐scale phase normalΦitalicMacro can reproduce different storage schemes. © 2023 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Gamma oscillations are thought to play a role in information processing in the brain. Bursting neurons, which exhibit periodic clusters of spiking activity, are a type of neuron that are thought to contribute largely to gamma oscillations. However, little is known about how the properties of bursting neurons affect the emergence of gamma oscillation, its waveforms, and its synchronized characteristics, especially when subjected to stochastic fluctuations. In this study, we proposed a bursting neuron model that can analyze the bursting ratio and the phase response function. Then we theoretically analyzed the neuronal population dynamics composed of bursting excitatory neurons, mixed with inhibitory neurons. The bifurcation analysis of the equivalent Fokker-Planck equation exhibits three types of gamma oscillations of unimodal firing, bimodal firing in the inhibitory population, and bimodal firing in the excitatory population under different interaction strengths. The analyses of the macroscopic phase response function by the adjoint method of the Fokker-Planck equation revealed that the inhibitory doublet facilitates synchronization of the high-frequency oscillations. When we keep the strength of interactions constant, decreasing the bursting ratio of the individual neurons increases the relative high-gamma component of the populational phase-coupling functions. This also improves the ability of the neuronal population model to synchronize with faster oscillatory input. The analytical frameworks in this study provide insight into nontrivial dynamics of the population of bursting neurons, which further suggest that bursting neurons have an important role in rhythmic activities.

Information processing in the brain is performed by the interactions of numerous number of neurons. However, much remains unknown about complex behaviors of a network composed of numerous number of neurons. Especially, in what case cross frequency coupling emerges and how it alters timings of individual firing are open questions. Therefore, in order to quantitatively evaluate the dynamics of the nervous group, we reproduced phenomena such as cross frequency coupling observed in the brain on a mathematical model and analyzed. The analyses of Fokker‐Planck equation provide the region of gamma/theta oscillations as well as their cross‐frequency couplings. In addition, we demonstrated that the populational oscillation synchronizes sequences of spike trains, which is supposed to be a possible mechanism of memory coding in hippocampus by theta‐gamma neural code.

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There are typically two distinct types of single‐neuron dynamics, known as Class I and Class II. Fast‐spiking (FS) neurons are a type of Class II neuron, which tend to exhibit periodic spikes in the gamma frequency. Although the population dynamics of Class I neurons are well elucidated with the Fokker‐Planck equation (FPE), much remains unknown about the population dynamics of Class II neurons. This is because the population model of Class II neurons typically leads to a high‐dimensional FPE that is hard to analyze. The modified theta (MT) model is one of the simplest mathematical models of Class I neurons, which possesses voltage‐dependent dynamics and has advantages in the analysis of population dynamics. In this study, we propose two approximation methods to derive a one‐dimensional FPE for FS interneurons using the framework of MT transformation. One method is mean‐field approximation of the variable related to adaptation; the other is semi‐adiabatic approximation of the variable. We confirm the firing characteristics of a single neuron in these approximated models match well to that of the original model proposed by Izhikevich. For the populational dynamics of the gamma oscillation, FPEs with these approximations make it possible to identify the region where Class II neuron firings synchronize. Bifurcation analyses of two FPEs show that semi‐adiabatic approximation fits better to the numerical results than mean‐field approximation. FPE with semi‐adiabatic approximation better illustrates the fact that a population of FS interneurons has a narrower region of gamma oscillation than that of Class I neurons. Moreover, analysis with semi‐adiabatic approximation of a population composed of both Class I and FS interneurons has a narrower region of synchronization with an increase in the FS interneuron ratio. © 2020 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.