A dynamical systems approach for estimating phase interactions between rhythms of different frequencies from experimental data.

Synchronization of neural oscillations as a mechanism of brain function is attracting increasing attention. Neural oscillation is a rhythmic neural activity that can be easily observed by noninvasive electroencephalography (EEG). Neural oscillations show the same frequency and cross-frequency synchronization for various cognitive and perceptual functions. However, it is unclear how this neural synchronization is achieved by a dynamical system. If neural oscillations are weakly coupled oscillators, the dynamics of neural synchronization can be described theoretically using a phase oscillator model. We propose an estimation method to identify the phase oscillator model from real data of cross-frequency synchronized activities. The proposed method can estimate the coupling function governing the properties of synchronization. Furthermore, we examine the reliability of the proposed method using time-series data obtained from numerical simulation and an electronic circuit experiment, and show that our method can estimate the coupling function correctly. Finally, we estimate the coupling function between EEG oscillation and the speech sound envelope, and discuss the validity of these results.

Evaluation of the Phase-Dependent Rhythm Control of Human Walking Using Phase Response Curves.

Humans and animals control their walking rhythms to maintain motion in a variable environment. The neural mechanism for controlling rhythm has been investigated in many studies using mechanical and electrical stimulation. However, quantitative evaluation of rhythm variation in response to perturbation at various timings has rarely been investigated. Such a characteristic of rhythm is described by the phase response curve (PRC). Dynamical simulations of human skeletal models with changing walking rhythms (phase reset) described a relation between the effective phase reset on stability and PRC, and phase reset around touch-down was shown to improve stability. A PRC of human walking was estimated by pulling the swing leg, but such perturbations hardly influenced the stance leg, so the relation between the PRC and walking events was difficult to discuss. This research thus examines human response to variations in floor velocity. Such perturbation yields another problem, in that the swing leg is indirectly (and weakly) perturbed, so the precision of PRC decreases. To solve this problem, this research adopts the weighted spike-triggered average (WSTA) method. In the WSTA method, a sequential pulsed perturbation is used for stimulation. This is in contrast with the conventional impulse method, which applies an intermittent impulsive perturbation. The WSTA method can be used to analyze responses to a large number of perturbations for each sequence. In the experiment, perturbations are applied to walking subjects by rapidly accelerating and decelerating a treadmill belt, and measured data are analyzed by the WSTA and impulse methods. The PRC obtained by the WSTA method had clear and stable waveforms with a higher temporal resolution than those obtained by the impulse method. By investigation of the rhythm transition for each phase of walking using the obtained PRC, a rhythm change that extends the touch-down and mid-single support phases is found to occur.

Interlimb coordination is not strictly controlled during walking.

In human walking, the left and right legs move alternately, half a stride out of phase with each other. Although various parameters, such as stride frequency and length, vary with walking speed, the antiphase relationship remains unchanged. In contrast, during walking in left-right asymmetric situations, the relative phase shifts from the antiphase condition to compensate for the asymmetry. Interlimb coordination is important for adaptive walking and we expect that interlimb coordination is strictly controlled during walking. However, the control mechanism remains unclear. In the present study, we derived a quantity that models the control of interlimb coordination during walking using two coupled oscillators based on the phase reduction theory and Bayesian inference method. The results were not what we expected. Specifically, we found that the relative phase is not actively controlled until the deviation from the antiphase condition exceeds a certain threshold. In other words, the control of interlimb coordination has a dead zone like that in the case of the steering wheel of an automobile. It is conjectured that such forgoing of control enhances energy efficiency and maneuverability. Our discovery of the dead zone in the control of interlimb coordination provides useful insight for understanding gait control in humans.

A mathematical model of negative covariability of inter-columnar excitatory synaptic actions caused by presynaptic inhibition.

We previously showed that a positive covariability between intracortical excitatory synaptic actions onto the two layer three pyramidal cells (PCs) located in mutually adjacent columns is changed into a negative covariability by column-wise presynaptic inhibition of intracortical inputs, implicated as a basis for the desynchronization of inter-columnar synaptic actions. Here we investigated how the inter-columnar desynchronization is modulated by the strength of presynaptic inhibition or other factors, by using a mathematical model. Based on our previous findings on the paired-pulse depression (PPD) of intracortical excitatory postsynaptic currents (EPSCs) evoked in PCs located in the stimulated home column (HC) but no PPD in PCs located in the adjacent column (AC), a mathematical model of synaptic connections between PCs and inhibitory interneurons was constructed. When the paired-pulse ratio (PPR) was decreased beyond 0.80, the correlation coefficient between the two second EPSC amplitudes in the paired PCs located in the HC and AC and that in the paired PCs located in the same HC exhibited opposite changes, and reached a global negative maximum and local positive maximum, respectively, at almost the same PPR (0.40). At this PPR, the desynchronization between the two cell assemblies in mutually adjacent columns would be maximized. These positive and negative covariabilities were not produced without background oscillatory synchronization across columns and were enhanced by increasing the synchronization magnitude, indicating that the synchronization leads to the desynchronization. We propose that a slow oscillatory synchronization across columns may emerge following the liberation from the column-wise presynaptic inhibition of inter-columnar synaptic inputs.

Scale-free structures emerging from co-evolution of a network and the distribution of a diffusive resource on it.

Co-evolution exhibited by a network system, involving the intricate interplay between the dynamics of the network itself and the subsystems connected by it, is a key concept for understanding the self-organized, flexible nature of real-world network systems. We propose a simple model of such coevolving network dynamics, in which the diffusion of a resource over a weighted network and the resource-driven evolution of the link weights occur simultaneously. We demonstrate that, under feasible conditions, the network robustly acquires scale-free characteristics in the asymptotic state. Interestingly, in the case that the system includes dissipation, it asymptotically realizes a dynamical phase characterized by an organized scale-free network, in which the ranking of each node with respect to the quantity of the resource possessed thereby changes ceaselessly. Our model offers a unified framework for understanding some real-world diffusion-driven network systems of diverse types.

Replicating receptive fields of simple and complex cells in primary visual cortex in a neuronal network model with temporal and population sparseness and reliability.

We propose a new principle for replicating receptive field properties of neurons in the primary visual cortex. We derive a learning rule for a feedforward network, which maintains a low firing rate for the output neurons (resulting in temporal sparseness) and allows only a small subset of the neurons in the network to fire at any given time (resulting in population sparseness). Our learning rule also sets the firing rates of the output neurons at each time step to near-maximum or near-minimum levels, resulting in neuronal reliability. The learning rule is simple enough to be written in spatially and temporally local forms. After the learning stage is performed using input image patches of natural scenes, output neurons in the model network are found to exhibit simple-cell-like receptive field properties. When the output of these simple-cell-like neurons are input to another model layer using the same learning rule, the second-layer output neurons after learning become less sensitive to the phase of gratings than the simple-cell-like input neurons. In particular, some of the second-layer output neurons become completely phase invariant, owing to the convergence of the connections from first-layer neurons with similar orientation selectivity to second-layer neurons in the model network. We examine the parameter dependencies of the receptive field properties of the model neurons after learning and discuss their biological implications. We also show that the localized learning rule is consistent with experimental results concerning neuronal plasticity and can replicate the receptive fields of simple and complex cells.

Multistable attractors in a network of phase oscillators with three-body interactions.

Three-body interactions have been found in physics, biology, and sociology. To investigate their effect on dynamical systems, as a first step, we study numerically and theoretically a system of phase oscillators with a three-body interaction. As a result, an infinite number of multistable synchronized states appear above a critical coupling strength, while a stable incoherent state always exists for any coupling strength. Owing to the infinite multistability, the degree of synchrony in an asymptotic state can vary continuously within some range depending on the initial phase pattern.

Interaction mechanisms quantified from dynamical features of frog choruses.

We employ a mathematical model (a phase oscillator model) to describe the deterministic and stochastic features of frog choruses in which male frogs attempt to avoid call overlaps. The mathematical model with a general interaction term is identified using a Bayesian approach, and it qualitatively reproduces the stationary and dynamical features of the empirical data. In addition, we quantify the magnitude of attention paid among the male frogs from the identified model, and then analyse the relationship between attention and behavioural parameters using a statistical approach. Our analysis demonstrates a negative correlation between attention and inter-frog distance, and also suggests a behavioural strategy in which male frogs selectively attend to a less attractive male frog (i.e. a male producing calls at longer intervals) in order to more effectively advertise their superior relative attractiveness to females.

Effect of recurrent infomax on the information processing capability of input-driven recurrent neural networks.

Reservoir computing is a framework for exploiting the inherent transient dynamics of recurrent neural networks (RNNs) as a computational resource. On the basis of this framework, much research has been conducted to evaluate the relationship between the dynamics of RNNs and the RNNs' information processing capability. In this study, we present a detailed analysis of the information processing capability of an RNN optimized by recurrent infomax (RI), an unsupervised learning method that maximizes the mutual information of RNNs by adjusting the connection weights of the network. The results indicate that RI leads to the emergence of a delay-line structure and that the network optimized by the RI possesses a superior short-term memory, which is the ability to store the temporal information of the input stream in its transient dynamics.

Optimal weighted networks of phase oscillators for synchronization.

The phase order parameter of oscillators on a network is optimized using two different sets of constraints. First, the maximization is achieved by adjusting the coupling strengths among the oscillators without changing the total coupling strength and the natural frequencies of the oscillators. This optimization reveals that a stronger weight tends to be assigned to a connection between two oscillators with greatly different natural frequencies. Second, we vary both coupling strengths and natural frequencies while maximizing the phase order and minimizing the penalty function which prevents the natural frequencies of the oscillators from taking the same value. This optimization reveals that a large total coupling strength makes oscillators take two natural frequencies (two-group state), whereas a small total coupling strength facilitates the convergence of natural frequencies to one single value (one-group state). Small and large penalty parameters make the optimized network take the one- and two-group states, respectively. This phase transition is observed in all-to-all, lattice, and scale-free networks although the clustering coefficient of the strongest links in the optimized network reflects the difference of the underlying network topologies.