Gouhei Tanaka

Complex-Valued Multistate Associative Memory With Nonlinear Multilevel Functions for Gray-Level Image Reconstruction

The complex-signum function has been widely used as an activation function in complex-valued recurrent neural networks for multistate associative memory. This paper presents two alternative activation functions with circularity. One is the complex-sigmoid function based on a multilevel sigmoid function defined on a circle. The other is a characteristic of a bifurcating neuron represented by a circle map. The performance of the complex-valued neural networks with the two kinds of activation functions is investigated in multistate associative memory tests. In both networks, the connection weights to store the memory patterns are determined by the generalized projection rule. We also demonstrate gray-level image reconstruction as a possible application of the proposed methods.

Complex-valued multistate associative memory with nonlinear multilevel functions for gray-level image reconstruction

A widely used complex-valued activation function for complex-valued multistate Hopfield networks is revealed to be essentially based on a multilevel step function. By replacing the multilevel step function with other multilevel characteristics, we present two alternative complex-valued activation functions. One is based on a multilevel sigmoid function, while the other on a characteristic of a multistate bifurcating neuron. Numerical experiments show that both modifications to the complex-valued activation function bring about improvements in network performance for a multistate associative memory. The advantage of the proposed networks over the complex-valued Hopfield networks with the multilevel step function is more outstanding when a complex-valued neuron represents a larger number of multivalued states. Further, the performance of the proposed networks in reconstructing noisy 256 gray-level images is demonstrated in comparison with other recent associative memories to clarify their advantages and disadvantages.

Mathematical modelling of prostate cancer growth and its application to hormone therapy

Hormone therapy in the form of androgen deprivation is a major treatment for advanced prostate cancer. However, if such therapy is overly prolonged, tumour cells may become resistant to this treatment and result in recurrent fatal disease. Long-term hormone deprivation also is associated with side effects poorly tolerated by patients. In contrast, intermittent hormone therapy with alternating on- and off-treatment periods is a possible clinical strategy to delay progression to hormone-refractory disease with the advantage of reduced side effects during the off-treatment periods. In this paper, we first overview previous studies on mathematical modelling of prostate tumour growth under intermittent hormone therapy. The model is categorized into a hybrid dynamical system because switching between on-treatment and off-treatment intervals is treated in addition to continuous dynamics of tumour growth. Next, we present an extended model of stochastic differential equations and examine how well the model is able to capture the characteristics of authentic serum prostate-specific antigen (PSA) data. We also highlight recent advances in time-series analysis and prediction of changes in serum PSA concentrations. Finally, we discuss practical issues to be considered towards establishment of mathematical model-based tailor-made medicine, which defines how to realize personalized hormone therapy for individual patients based on monitored serum PSA levels.

Synchronization and propagation of bursts in networks of coupled map neurons.

The present paper studies regular and complex spatiotemporal behaviors in networks of coupled map-based bursting oscillators. In-phase and antiphase synchronization of bursts are studied, explaining their underlying mechanisms in order to determine how network parameters separate them. Conditions for emergent bursting in the coupled system are derived from our analysis. In the region of emergence, patterns of chaotic transitions between synchronization and propagation of bursts are found. We show that they consist of transient standing and rotating waves induced by symmetry-breaking bifurcations, and can be viewed as a manifestation of the phenomenon of chaotic itinerancy.

Dynamical robustness in complex networks: the crucial role of low-degree nodes

Many social, biological, and technological networks consist of a small number of highly connected components (hubs) and a very large number of loosely connected components (low-degree nodes). It has been commonly recognized that such heterogeneously connected networks are extremely vulnerable to the failure of hubs in terms of structural robustness of complex networks. However, little is known about dynamical robustness, which refers to the ability of a network to maintain its dynamical activity against local perturbations. Here we demonstrate that, in contrast to the structural fragility, the nonlinear dynamics of heterogeneously connected networks can be highly vulnerable to the failure of low-degree nodes. The crucial role of low-degree nodes results from dynamical processes where normal (active) units compensate for the failure of neighboring (inactive) units at the expense of a reduction in their own activity. Our finding highlights the significant difference between structural and dynamical robustness in complex networks.

Random and Targeted Interventions for Epidemic Control in Metapopulation Models

In general, different countries and communities respond to epidemics in accordance with their own control plans and protocols. However, owing to global human migration and mobility, strategic planning for epidemic control measures through the collaboration of relevant public health administrations is gaining importance for mitigating and containing large-scale epidemics. Here, we present a framework to evaluate the effectiveness of random (non-strategic) and targeted (strategic) epidemic interventions for spatially separated patches in metapopulation models. For a random intervention, we analytically derive the critical fraction of patches that receive epidemic interventions, above which epidemics are successfully contained. The analysis shows that the heterogeneity of patch connectivity makes it difficult to contain epidemics under the random intervention. We demonstrate that, particularly in such heterogeneously connected networks, targeted interventions are considerably effective compared to the random intervention. Our framework is useful for identifying the target areas where epidemic control measures should be focused.

Mathematically modelling and controlling prostate cancer under intermittent hormone therapy.

In this review, we summarize our recently developed mathematical models that predict the effects of intermittent androgen suppression therapy on prostate cancer (PCa). Although hormone therapy for PCa shows remarkable results at the beginning of treatment, cancer cells frequently acquire the ability to grow without androgens during long-term therapy, resulting in an eventual relapse. To circumvent hormone resistance, intermittent androgen suppression was investigated as an alternative treatment option. However, at the present time, it is not possible to select an optimal schedule of on- and off-treatment cycles for any given patient. In addition, clinical trials have revealed that intermittent androgen suppression is effective for some patients but not for others. To resolve these two problems, we have developed mathematical models for PCa under intermittent androgen suppression. The mathematical models not only explain the mechanisms of intermittent androgen suppression but also provide an optimal treatment schedule for the on- and off-treatment periods.

MULTISTATE ASSOCIATIVE MEMORY WITH PARAMETRICALLY COUPLED MAP NETWORKS

The present paper proposes two types of parametrically coupled circle map networks for multistate associative memory. One of the networks uses a circle map exhibiting an attractor-merging crisis of multiple chaotic attractors to represent a multistate element. The other uses another circle map whose bifurcation diagram serves as a substitute for a multilevel activation function. The configuration of each network is suitably selected according to the dynamics of the individual circle map so that the network can bring about self-organizing chaotic dynamics with an association of a memory. Namely, the coupling term is determined by the generalized partial error function in the first network, and by the weighted sum of inputs in the second network. These multistate networks can be considered as extensions of two kinds of interesting binary networks called the parametrically coupled sine map networks [Lee & Farhat, 2001a], respectively. We illustrate that the proposed networks can exhibit desirable associative dynamics that is missing in the conventional multistate networks.

BIFURCATION STRUCTURES OF PERIOD-ADDING PHENOMENA IN AN OCEAN INTERNAL WAVE MODEL

In this paper, we study bifurcation structures of period-adding phenomena in an internal wave model that is a mathematical model for ocean internal waves. It has been suggested that chaotic solutions observed in the internal wave model may be related to the universal property of the energy spectra of ocean internal waves. In numerical bifurcation analyses of the internal wave model, we illustrate bifurcation routes to chaos and parameter regions where chaotic behavior is observed. Furthermore, it is found that the chaotic solutions are related to the period-adding sequence, that is, successive generations of periodic solutions with longer periods as a control parameter is changed. Considering the period-adding sequence as successive local bifurcations, we discuss a mechanism of the phenomena from the viewpoint of bifurcation analysis. We also consider similarity between period-adding phenomena in the internal wave model and ones in the Lorenz model.

Wave-Based Reservoir Computing by Synchronization of Coupled Oscillators

We propose wave-based computing based on coupled oscillators to avoid the inter-connection bottleneck in large scale and densely integrated cognitive systems. In addition, we introduce the concept of reservoir computing to coupled oscillator systems for non-conventional physical implementation and reduction of the training cost of large and dense cognitive systems. We show that functional approximation and regression can be efficiently performed by synchronization of coupled oscillators and subsequent simple readouts.