Kosuke Hamaguchi

Dynamics and computation of continuous attractors

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.

Correlated Firing in a Feedforward Network with Mexican-Hat-Type Connectivity

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.

Variable Timescales of Repeated Spike Patterns in Synfire Chain with Mexican-Hat Connectivity

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.

Local excitation solutions in one-dimensional neural fields by external input stimuli

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.

Theory of Interaction of Memory Patterns in Layered Associative Networks

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 th...

Stochastic resonance of localized activity driven by common noise

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.

Sparse and Dense Encoding in Layered Associative Network of Spiking Neurons

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) 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 sh...

The Tracking Speed of Continuous Attractors

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.

Stochasticity in localized synfire chain

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 pre-synaptic 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.