We examine a previouly introduced attractor neural network model that explains the persistent activities of neurons in the anterior ventral temporal cortex of the brain. In this model, the coexistence of several attractors including correlated attractors was reported in the cases of finite and infinite loading. In this paper, by means of a statistical mechanical method, we study the statics and dynamics of the model in both finite and extensive loading, mainly focusing on the retrieval properties of the Hopfield and correlated attractors. In the extensive loading case, we derive the evolution equations by the dynamical replica theory. We found several characteristic temporal behaviours, both in the finite and extensive loading cases. The theoretical results were confirmed by numerical simulations.

We study the problem of performance evaluation of code-division multiple-access (CDMA) multiuser detectors by means of statistical mechanics using the replica method. The replica symmetric solutions were analyzed previously. As is well known, the replica symmetric solution becomes unstable when the temperature (magnitude of noise) is lowered. In this paper, we investigate both the behavior and the stability of the replica symmetric solutions in the low temperature region. We find that the solutions have complicated bifurcation structures in the low temperature region where the solutions coexist. We also find that there are two types of replica symmetry breaking, Almeida-Thouless (AT)-instability and freezing. We obtain the one-step replica symmetry breaking solution in each case. Further, we compare the theoretical results with the results from the Monte-Carlo simulations. Consequently, we find that the theoretical results agree with the numerical simulations.

Abstract. We study self-programming in recurrent neural networks where both neurons (the 'processors') and synaptic interactions ('the programme') evolve in time simultaneously, according to specific coupled stochastic equations.The interactions are divided into a hierarchy of L groups with adiabatically separated and monotonically increasing time-scales, representing sub-routines of the system programme of decreasing volatility. We solve this model in equilibrium, assuming ergodicity at every level, and find as our replica-symmetric solution a formalism with a structure similar but not identical to Parisi's L-step replica symmetry breaking scheme. Apart from differences in details of the equations (due to the fact that here interactions, rather than spins, are grouped into clusters with different time-scales), in the present model the block sizes m i of the emerging ultrametric solution are not restricted to the interval [0, 1], but are independent control parameters, defined in terms of the noise strengths of the various levels in the hierarchy, which can take any value in [0, ∞ . This is shown to lead to extremely rich phase diagrams, with an abundance of firstorder transitions especially when the level of stochasticity in the interaction dynamics is chosen to be low.

We study a model of an idiotypic immune network which was introduced by N. K. Jerne. It is known that in immune systems there generally exist several kinds of immune cells which can recognize any particular antigen. Taking this fact into account and assuming that each cell interacts with only a finite number of other cells, we analyze a large scale immune network via both numerical simulations and statistical mechanical methods, and show that the distribution of the concentrations of antibodies becomes non-trivial for a range of values of the strength of the interaction and the connectivity. §1. IntroductionThere are many numerical and theoretical studies of biological networks in which there is a global coupling between the constituent elements, e.g. Hopfield-type neural networks and networks of Kuramoto-type phase oscillators. 1), 2) In contrast, relatively few studies have been carried out of networks where each component interacts with only a small number of randomly selected other components. One such system is the immune network, introduced by Jerne 3) to explain the activation of immune cells in the absence of external stimulation.Let us briefly summarize the main mechanism of cell-interaction in the immune system. Its main constituents are B-lymphocytes (B-cells), T-lymphocytes (T-Cells) and antibodies produced by B-cells. B-cells and T-cells have receptors on their surfaces. The receptors of B-cells are antibodies, which recognize and connect to antigens in order to neutralize them; they have specific 3-dimensional structures which are called 'idiotypes'. A family of B-cells which are generated from a given B-cell is called a 'clone'. Hence, all members of a clone as well as the antibodies produced by this clone have the same idiotype. In general, each antibody could present several 3-dimensional structures which can be recognized by other B-cells. This then generates, indirectly, an effective interaction between antibodies. This paper is organized as follows. In §2 we formulate the model, followed by a summary of previous studies in §3. In §4, we present the results of our present study. Section 5 is devoted to a summary and a discussion.

We analyze the on-line learning of a Perceptron from signals produced by a single Perceptron suffering from external noise or by two independent Perceptrons without noise. We adopt typical three learning rules in both single-teacher and two-teacher cases. For the single-teacher case, we treat the input and output noises and for the two-teacher case, we assume that signals are given by two teachers with a definite probability. In the single-teacher case, in order to improve the learning when it does not succeed in the sense that the student vector does not converge to the teacher vector, we use two methods: a method based on the optimal learning rate and an averaging method. Furthermore, we obtain an asymptotic form of the generalization error using an optimal learning rate for the three learning rules, and we estimate noise parameters using the simulation data by the averaging method. In the two-teacher case, for the Hebbian rule, we give analytical solutions of order parameters. Furthermore, we estimate noise parameters using the Perceptron rule by the averaging method. The theoretical results agree quite well with the numerical simulations.