Language：English   Publishing type：Research paper (other academic)   Publisher：Cold Spring Harbor Laboratory  

To optimize decision making, animals need to execute not only a strategy to choose a good option but sometimes also one to avoid a bad option. A psychological study indicates that positive and negative information is processed in a different manner in the brain. The nucleus accumbens (NAc) contains two different types of neurons, dopamine D1 and D2 receptor-expressing neurons which are implicated in reward-based decision making and aversive learning. However, little is known about the neural mechanisms by which D1 or D2 receptor-expressing neurons in the NAc contribute to the execution of the strategy to choose a good option or one to avoid a bad option under decision making. Here, we have developed two novel visual discrimination tasks for mice to assess the strategy to choose a good option and one to avoid a bad option. By chemogenetically suppressing the subpopulation of the NAc neurons, we have shown that dopamine D2 receptor-expressing neurons in the NAc selectively contribute to the strategy to avoid a bad option under reward-based decision making. Furthermore, our optogenetic and calcium imaging experiments indicate that dopamine D2 receptor-expressing neurons are activated by error choices and the activation following an error plays an important role in optimizing the strategy in the next trial. Our findings suggest that the activation of D2 receptor-expressing neurons by error choices through learning enables animals to execute the appropriate strategy.

DOI： 10.1101/2021.08.05.455353

Language：English   Publishing type：Research paper (scientific journal)  

The subthalamic nucleus (STN), a key component of the basal ganglia circuitry, receives inputs from broad cerebral cortical areas and relays cortical activity to subcortical structures. Recent human and animal studies have suggested that executive function, which is assumed to consist of a set of different cognitive processes for controlling behavior, depends on precise information processing between the cerebral cortex and subcortical structures, leading to the idea that the STN contains neurons that transmit the information required for cognitive processing through their activity, and is involved in such cognitive control directly and dynamically. On the other hand, the STN activity also affects intracellular signal transduction and gene expression profiles influencing plasticity in other basal ganglia components. The STN may also indirectly contribute to information processing for cognitive control in other brain areas by regulating slower signaling mechanisms. However, the precise correspondence and causal relationship between the STN activity and cognitive processes are not fully understood. To address how the STN activity is involved in cognitive processes for controlling behavior, we applied Designer Receptors Exclusively Activated by Designer Drugs (DREADD)-based chemogenetic manipulation of neural activity to behavioral analysis using a touchscreen operant platform. We subjected mice selectively expressing DREADD receptors in the STN neurons to a five-choice serial reaction time task, which has been developed to quantitatively measure executive function. Chemogenetic suppression of the STN activity reversibly impaired attention, especially required under highly demanding conditions, and increased impulsivity but not compulsivity. These findings, taken together with the results of previous lesion studies, suggest that the STN activity, directly and indirectly, participates in cognitive processing for controlling behavior, and dynamically regulates specific types of subprocesses in cognitive control probably through fast synaptic transmission.

DOI： 10.3389/fnsys.2020.00038

PubMed

Language：Japanese   Publisher：Japanese Society for Medical and Biological Engineering  

Sequences of neural activity are thought to play an important role in motor control. The neural mechanisms that give rise to these sequences are not well understood, but an influential idea is that activity propagation in ensembles of neurons can generate sequential activity (i.e., a synfire chain). Birdsong is an elaborate and stereotyped vocal behavior controlled with millisecond precision, and various lines of evidence support the hypothesis that song premotor neurons located in a telencephalic nucleus HVC form a synaptic chain to generate song tempo. Here we combine brain temperature manipulation, synaptic activity recording, and computational methods to show that song tempo is not generated by a local mechanism of HVC but instead is the product of a distributed and recurrent synaptic network spanning the forebrain and brainstem. Using a miniature Peltier device, we found that focally manipulating the temperature of HVC exerted much greater effect on activity propagation locally within HVC than it did on song tempo, however, exerted identical effects on song tempo and activity propagation through a recurrent network that contains HVC as one of its elements. The potential models that can account for the statistical structure of synaptic timing distribution of HVC neurons will be discussed in the talk.

DOI： 10.11239/jsmbe.53.S98_02

Language：English   Publishing type：Research paper (scientific journal)   Publisher：PUBLIC LIBRARY SCIENCE  

The computational complexity of the brain depends in part on a neuron's capacity to integrate electrochemical information from vast numbers of synaptic inputs. The measurements of synaptic activity that are crucial for mechanistic understanding of brain function are also challenging, because they require intracellular recording methods to detect and resolve millivolt-scale synaptic potentials. Although glass electrodes are widely used for intracellular recordings, novel electrodes with superior mechanical and electrical properties are desirable, because they could extend intracellular recording methods to challenging environments, including long term recordings in freely behaving animals. Carbon nanotubes (CNTs) can theoretically deliver this advance, but the difficulty of assembling CNTs has limited their application to a coating layer or assembly on a planar substrate, resulting in electrodes that are more suitable for in vivo extracellular recording or extracellular recording from isolated cells. Here we show that a novel, yet remarkably simple, millimeter-long electrode with a sub-micron tip, fabricated from self-entangled pure CNTs can be used to obtain intracellular and extracellular recordings from vertebrate neurons in vitro and in vivo. This fabrication technology provides a new method for assembling intracellular electrodes from CNTs, affording a promising opportunity to harness nanotechnology for neuroscience applications.

DOI： 10.1371/journal.pone.0065715

Web of Science

PubMed

Language：English   Publishing type：Research paper (scientific journal)   Publisher：SOC NEUROSCIENCE  

Complex brain functions, such as the capacity to learn and modulate vocal sequences, depend on activity propagation in highly distributed neural networks. To explore the synaptic basis of activity propagation in such networks, we made dual in vivo intracellular recordings in anesthetized zebra finches from the input (nucleus HVC, used here as a proper name) and output [lateral magnocellular nucleus of the anterior nidopallium (LMAN)] neurons of a songbird cortico-basal ganglia (BG) pathway necessary to the learning and modulation of vocal motor sequences. These recordings reveal evidence of bidirectional interactions, rather than only feedforward propagation of activity from HVC to LMAN, as had been previously supposed. A combination of dual and triple recording configurations and pharmacological manipulations was used to map out circuitry by which activity propagates from LMAN to HVC. These experiments indicate that activity travels to HVC through at least two independent ipsilateral pathways, one of which involves fast signaling through a midbrain dopaminergic cell group, reminiscent of recurrent mesocortical loops described in mammals. We then used in vivo pharmacological manipulations to establish that augmented LMAN activity is sufficient to restore high levels of sequence variability in adult birds, suggesting that recurrent interactions through highly distributed forebrain-midbrain pathways can modulate learned vocal sequences.

DOI： 10.1523/JNEUROSCI.1666-12.2012

Web of Science

PubMed

Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER PHYSIOLOGICAL SOC  

Hamaguchi K, Riehle A, Brunel N. Estimating network parameters from combined dynamics of firing rate and irregularity of single neurons. J Neurophysiol 105: 487-500, 2011. First published August 18, 2010; doi:10.1152/jn.00858.2009. High firing irregularity is a hallmark of cortical neurons in vivo, and modeling studies suggest a balance of excitation and inhibition is necessary to explain this high irregularity. Such a balance must be generated, at least partly, from local interconnected networks of excitatory and inhibitory neurons, but the details of the local network structure are largely unknown. The dynamics of the neural activity depends on the local network structure; this in turn suggests the possibility of estimating network structure from the dynamics of the firing statistics. Here we report a new method to estimate properties of the local cortical network from the instantaneous firing rate and irregularity (CV2) under the assumption that recorded neurons are a part of a randomly connected sparse network. The firing irregularity, measured in monkey motor cortex, exhibits two features; many neurons show relatively stable firing irregularity in time and across different task conditions; the time-averaged CV2 is widely distributed from quasi-regular to irregular (CV2 = 0.3-1.0). For each recorded neuron, we estimate the three parameters of a local network [balance of local excitation-inhibition, number of recurrent connections per neuron, and excitatory postsynaptic potential (EPSP) size] that best describe the dynamics of the measured firing rates and irregularities. Our analysis shows that optimal parameter sets form a two-dimensional manifold in the three-dimensional parameter space that is confined for most of the neurons to the inhibition-dominated region. High irregularity neurons tend to be more strongly connected to the local network, either in terms of larger EPSP and inhibitory PSP size or larger number of recurrent connections, compared with the low irregularity neurons, for a given excitatory/inhibitory balance. Incorporating either synaptic short-term depression or conductance-based synapses leads many low CV2 neurons to move to the excitation-dominated region as well as to an increase of EPSP size.

DOI： 10.1152/jn.00858.2009

Web of Science

PubMed

Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

Cortical neurons are massively connected with other cortical and subcortical cells, and they receive synaptic inputs from multiple sources. To explore the basis of how interconnected cortical cells are locally activated by such inputs, we theoretically analyze the local excitation patterns elicited by external input stimuli by using a one-dimensional neural field model. We examine the conditions for the existence and stability of the local excitation solutions under arbitrary time-invariant inputs and establish a graphic analysis method that can detect all steady local excitation solutions and examine their stability. We apply this method to a case where a pair of supra- and subthreshold stimuli are applied to nearby positions in the field. The results demonstrate that there can exist bistable local excitation solutions with different lengths and that the local excitation exhibits hysteretic behavior when the relative distance between the two stimuli is altered.

DOI： 10.1007/s00521-009-0246-2

Web of Science

Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER PHYSICAL SOC  

The properties of solid states, biophysical materials, neuronal circuits, and equilibrium states of a many-body system can be studied by using techniques in statistical physics. It has been common practice to represent a system composed of binary state units by using an Ising spin network where each unit has symmetric {-1,1} states. However, the asymmetry or symmetry of the binary states of the units can affect the property and ergodicity of the system, but better understanding of the quantitative difference is still needed. We compare systems of binary units with symmetric or asymmetric states. The network has spatially modulated interaction with quenched randomness. We can bridge the Ising spin network and McCulloch-Pitts neuron network and analyze the stability of the system via replica method by introducing an interpolating parameter. The effects of the asymmetry states affect the multistability of the system and the stability of replica-symmetry solutions.

DOI： 10.1103/PhysRevE.78.051124

Web of Science

Language：English   Publishing type：Research paper (scientific journal)   Publisher：M I T PRESS  

Continuous attractor is a promising model for describing the encoding of continuous stimuli in neural systems. In a continuous attractor, the stationary states of the neural system form a continuous parameter space, on which the system is neutrally stable. This property enables the neutral system to track time-varying stimuli smoothly, but it also degrades the accuracy of information retrieval, since these stationary states are easily disturbed by external noise. In this work, based on a simple model, we systematically investigate the dynamics and the computational properties of continuous attractors. In order to analyze the dynamics of a large-size network, which is otherwise extremely complicated, we develop a strategy to reduce its dimensionality by utilizing the fact that a continuous attractor can eliminate the noise components perpendicular to the attractor space very quickly. We therefore project the network dynamics onto the tangent of the attractor space and simplify it successfully as a one-dimensional Ornstein-Uhlenbeck process. Based on this simplified model, we investigate (1) the decoding error of a continuous attractor under the driving of external noisy inputs, (2) the tracking speed of a continuous attractor when external stimulus experiences abrupt changes, (3) the neural correlation structure associated with the specific dynamics of a continuous attractor, and (4) the consequence of asymmetric neural correlation on statistical population decoding. The potential implications of these results on our understanding of neural information processing are also discussed.

DOI： 10.1162/neco.2008.10-06-378

Web of Science

PubMed

Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOC JAPAN  

A synfire chain is a simple neural network model which can transmit stable synchronous spikes called a pulse packet. However how synfire chains coexist in one network remains to be elucidated. We have studied the activity of a layered associative network of leaky integrate-and-fire neurons which connections are embedded with memory patterns by the Hebbian learning rule. We analyze their activity by the Fokker-Planck method. In our previous report, when a half of neurons belongs to each memory pattern (pattern rate F = 0.5), the temporal profiles of the network activity is split into temporally clustered groups called sublattices under certain input conditions. In this study, we show that when the network is sparsely connected (F < 0.5), synchronous firings of the memory pattern are promoted. On the contrary, the densely connected network (F > 0.5) inhibit synchronous firings. The basin of attraction and the storage capacity of the embedded memory patterns also depend on the sparseness of the network. We show that the sparsely (densely) connected networks enlarge (shrink) the basion of attraction and increase (decrease) the storage capacity.

DOI： 10.1143/JPSJ.76.124801

Web of Science

Language：English   Publishing type：Research paper (scientific journal)   Publisher：M I T PRESS  

Repetitions of precise spike patterns observed both in vivo and in vitro have been reported for more than a decade. Studies on the spike volley (a pulse packet) propagating through a homogeneous feedforward network have demonstrated its capability of generating spike patterns with millisecond fidelity. This model is called the synfire chain and suggests a possible mechanism for generating repeated spike patterns (RSPs). The propagation speed of the pulse packet determines the temporal property of RSPs. However, the relationship between propagation speed and network structure is not well understood. We studied a feedforward network with Mexican-hat connectivity by using the leaky integrate-and-fire neuron model and analyzed the network dynamics with the Fokker-Planck equation. We examined the effect of the spatial pattern of pulse packets on RSPs in the network with multistability. Pulse packets can take spatially uniform or localized shapes in a multistable regime, and they propagate with different speeds. These distinct pulse packets generate RSPs with different timescales, but the order of spikes and the ratios between interspike intervals are preserved. This result indicates that the RSPs can be transformed into the same template pattern through the expanding or contracting operation of the timescale.

DOI： 10.1162/neco.2007.19.9.2468

Web of Science

PubMed

Language：English   Publishing type：Research paper (international conference proceedings)   Publisher：SPRINGER-VERLAG BERLIN  

Continuous attractor is a promising model for describing the encoding of continuous stimuli in neural systems. In a continuous attractor, the stationary states of the neural system form a continuous parameter space, on which the system is neutrally stable. This property enables the neutral system to track time-varying stimulus smoothly. In this study we investigate the tracking speed of continuous attractors. In order to analyze the dynamics of a large-size network, which is otherwise extremely complicated, we develop a strategy to reduce its dimensionality by utilizing the fact that a continuous attractor can eliminate the input components perpendicular to the attractor space very quickly. We therefore project the network dynamics onto the tangent of the attractor space, and simplify it to be a one-dimension Ornstein-Uhlenbeck process. With this approximation we elucidate that the reaction time of a continuous attractor increases logarithmically with the size of the stimulus change. This finding may have important implication on the mental rotation behavior.

Web of Science

Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOC JAPAN  

A synfire chain is a network that can generate repeated spike patterns with millisecond precision. Although synfire chains with only one activity propagation mode have been intensively analyzed with several neuron models, those with several stable propagation modes have not been thoroughly investigated. By using the leaky integrate-and-fire neuron model, we constructed a layered associative network embedded with memory patterns. We analyzed the network dynamics with the Fokker-Planck equation. First, we addressed the stability of one memory pattern as a propagating spike volley. We showed that memory patterns propagate as pulse packets. Second, we investigated the activity when we activated two different memory patterns. Simultaneous activation of two memory patterns with the same strength led the propagating pattern to a mixed state. In contrast, when the activations had different strengths, the pulse packet converged to a two-peak state. Finally, we studied the effect of the preceding pulse packet on the following pulse packet. The following pulse packet was modified from its original activated memory pattern, and it converged to a two-peak state, mixed state or non-spike state depending on the time interval.

DOI： 10.1143/JPSJ.75.114803

Web of Science

Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMERICAN PHYSICAL SOC  

The replica method has played a key role in analyzing systems with disorder, e.g., the Sherrington-Kirkpatrick (SK) model, and associative neural networks. Here we study the influence of disorder in the lateral inhibition type interactions on the cooperative and uncooperative behavior of recurrent neural networks by using the replica method. Although the interaction between neurons has a dependency on distance, our model can be solved analytically. Bifurcation analysis identifies the boundaries between paramagnetic, ferromagnetic, spin-glass, and localized phases. In the localized phase, the network shows a bump like activity, which is often used as a model of spatial working memory or columnar activity in the visual cortex. Simulation results show that disordered interactions can stabilize the drift the of bump position, which is commonly observed in conventional lateral inhibition type neural networks.

DOI： 10.1103/PhysRevE.73.051104

Web of Science

Language：English   Publishing type：Research paper (scientific journal)   Publisher：MIT PRESS  

We report on deterministic and stochastic evolutions of firing states through a feedforward neural network with Mexican-hat-type connectivity. The prevalence of columnar structures in a cortex implies spatially localized connectivity between neural pools. Although feedforward neural network models with homogeneous connectivity have been intensively studied within the context of the synfire chain, the effect of local connectivity has not yet been studied so thoroughly. When a neuron fires independently, the dynamics of macroscopic state variables (a firing rate and spatial eccentricity of a firing pattern) is deterministic from the law of large numbers. Possible stable firing states, which are derived from deterministic evolution equations, are uniform, localized, and nonfiring. The multistability of these three states is obtained where the excitatory and inhibitory interactions among neurons are balanced. When the presynapse-dependent variance in connection efficacies is incorporated into the network, the variance generates common noise. Then the evolution of the macroscopic state variables becomes stochastic, and neurons begin to fire in a correlated manner due to the common noise. The correlation structure that is generated by common noise exhibits a nontrivial bimodal distribution. The development of a firing state through neural layers does not converge to a certain fixed point but keeps on fluctuating.

DOI： 10.1162/0899766054322937

Web of Science

PubMed

Language：English   Publishing type：Research paper (scientific journal)   Publisher：6  

We study the influence of spatially correlated noise on the transient dynamics of a recurrent network with Mexican-Hat-type connectivity. We derive the closed form of the order parameter functional in the thermodynamical limit of neuron number N. Our analysis shows that network dynamics is qualitatively changed by the presence of common noise. Network dynamics driven by common noise obtains the global level of fluctuation, which is not observed in a network driven by independent noise only. We show that the optimal level of global fluctuation enhances the transition from non-localized firing states to spatially localized firing states, and also enhances the rotation speed of localized activity. © Springer-Verlag 2005.

DOI： 10.1007/s00422-005-0570-2

Scopus

PubMed

Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

We report on stochastic evolutions of firing states through feedforward neural networks with Mexican-Hat-type connectivity. The variance in connectivity, which depends on the presynaptic neuron, generates a common noisy input to post-synaptic neurons. We develop a theory to describe the stochastic evolution of the localized synfire chain driven by a common noisy input. The development of a firing state through neural layers does not converge to a certain fixed point but keeps on fluctuating. Stationary firing states except for a non-firing state are lost, but an almost stationary distribution of firing state is observed. (c) 2004 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.neucom.2004.10.044

Web of Science

Language：Japanese   Publisher：The Institute of Electronics, Information and Communication Engineers  

CiNii Books

Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

A feedforward network with homogeneous connectivity cannot transmit quantitative information by one spike volley. In this paper, quantitative information transmission through neural layers connected by Mexican-Hat-type connectivity is examined. It is shown that the intensity of an input signal can be encoded as a size of an active region in a neural layer. (C) 2004 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.neucom.2004.01.027

Web of Science

Other Link： http://dblp.uni-trier.de/db/conf/nips/nips2004.html#conf/nips/HamaguchiOA04

Language：English   Publisher：The Physical Society of Japan (JPS)  

CiNii Books

Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

CiNii Books

Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

CiNii Books

Language：Japanese   Publisher：The Institute of Electronics, Information and Communication Engineers  

In the statistical mechanical framework, the firing and non-firing states of neuron are usually represented by the Ising spin (±1). On the other hand, they represented by 1 and 0 in the McCulloch-Pitts neuron. We compared the Ising spin and McCulloch-Pitts neuron systems which have random iteraction obeying the Gaussian distribution through the replica method. We found that it is imposible to define boudary between the paramagnetic and spin glass phases, and that the standard recipe of the AT stability can not applied in the McCulloch-Pitts neuron system.

CiNii Books

Language：Japanese   Publisher：The Institute of Electronics, Information and Communication Engineers  

A synfire chain is one of the networks which generate stable synchronous pulse packets. Although the networks with a single stable synfire state is intensively analyzed, the networks with several stable synfire states have not yet been investigated so thoroughly. By using leaky integrate-and-fire neuron model we construct a layered associative feedforward network embedded with several memory patterns. We analyze the network dynamics with the Fokker-Planck equation. First, we show that in the network the memory patterns propagate as pulse packets. Second, we find several characteristic phenomena, not observed in the conventional synfire chain.

CiNii Books

Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

CiNii Books

Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

CiNii Books

Language：English   Publishing type：Research paper, summary (international conference)   Publisher：ELSEVIER IRELAND LTD  

Web of Science

Language：Japanese   Publisher：The Institute of Electronics, Information and Communication Engineers  

Electrophysiological experiments of working memory task in the presence of intervening stimuli has shown that the delay activity in the prefrontal cortex (PFC) is preserved even after the intervening stimulus presentations, while that in the inferior temporal cortex (ITC) can be easily disturbed. In this study, we reproduce this type of working memory-related neuronal dynamics in PFC and ITC by constructing the two-layer neural field model. We also analyze the network pattern for mation underlying the memory process to discuss the conditions for reproducing the neuronal response observed in the task.

CiNii Books

Language：Japanese   Publisher：The Institute of Electronics, Information and Communication Engineers  

Pattern formation in a neural field model is theoretically analyzed. We show the conditions for existence and stability of local excitation solutions with arbitrary stationary input conditions to develop a graphic analysis method. The method is useful for finding all the local excitation solutions and their stability. By applying the method to a input condition where two input stimuli are applied, we show the existence of bistable pattern dynamics.

CiNii Books

Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

CiNii Books

Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

CiNii Books

Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

CiNii Books

Language：Japanese   Publisher：The Institute of Electronics, Information and Communication Engineers  

Stochastic resonance (SR) is the phenomena that the response of a non-linear system to a weak signal can be maximized by the presence of an appropriate level of noise. In the excitable system like a neuron, SR has been studied in single neuron, array, and lattice of them, but neural network models with functions, such as associative memory, and working memory, has not been studied so thoroughly. The macroscopic variables, or order parameters, obey deterministic dynamics in the presence of independent noise, however, the system shows stochastic dynamics when spatially correlated noise (common noise) is applied. Therefore, SR in a neural network driven by independent noise and that of common noise would be intrinsically different. In the present paper, by using the model of working memory where neurons are connected by local excitation and lateral inhibition, we show that the rotation speed of a localized activity is maximized by particular level of common noise and independent noise.

CiNii Books

Language：English   Publisher：ELSEVIER SCIENCE BV  

A feedforward network with homogeneous connectivity cannot transmit quantitative information by one spike volley. In this paper, quantitative information transmission through neural layers connected by Mexican-Hat-type connectivity is examined. It is shown that the intensity of an input signal can be encoded as a size of an active region in a neural layer. (C) 2004 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.neucom.2004.01.027

Web of Science

Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

CiNii Books

Language：Japanese   Publisher：The Institute of Electronics, Information and Communication Engineers  

It has been suggested for decades that information is encoded not only the mean firing rate of one neuron but also on the synchrony of neural spikes among the neuron population. Synfire chain is a model of homogeneous propagation of neuronal activity in a feedforward neural network. This homogeneity of the connection enables us to solve the dynamics of the network easily, but a real brain has inhomogeneous connectivity with in the neural network. Therefore, we propose a network model with Mexican-Hat type connectivity between the layers. By using McCulloch-Pitts model, we derive order-parameter equations, which describes the Fourier 0-th mode and 2-th mode concerning the propagating neuronal activity in the feedforward neural network. We found that two non-trivial propagation mode, i.e., localized activity and uniform activity, are stable as well as the trivial solution of non-firing state. We also found the bi-stable phase where the localized activity and the uniform activity coexist.

CiNii Books