Narrow Results

In this paper we discuss a noisy mean field model for the genetic toggle switch. We show that this model approximates very well the characteristics of the system, observed using the exact Gillespie stochastic simulation algorithm. Also, we show that the system can be made exponentially stable depending on reaction parameters.

Tilt angle (order parameter) and the susceptibility are calculated as a function of temperature for the $\alpha$–$\beta$ transition in quartz using a Landau phenomenological model. The tilt angle as obtained from the model is fitted to the experimental data from the literature and the temperature dependence of the tilt angle susceptibility is predicted close to the $\alpha$–$\beta$ transition in quartz. Our results show that the mean field model explains the observed behavior of the $\alpha$–$\beta$ phase transition in quartz adequately and it can be applied to some related materials.

This study investigates the stability of sparsely encoded associative memory in a network composed of stochastic neurons. The incorporation of short-term synaptic dynamics significantly changes the stability with respect to synaptic properties. Various states including static and oscillatory states are found in the network dynamics. Specifically, the sparseness of memory patterns raises the problem of spurious states. A mean field model is used to analyze the detailed structure in the stability and show that the performance of memory retrieval is recovered by appropriate feedback.

Generalized periodic discharges (GPD) are generalized waveforms that recur with a relatively uniform morphology and duration observed in EEG recordings of many types of metabolic encephalopathy, which are often referred to as epileptiform. In this paper, we try to link these spatiotemporal electrocortical activities and significant attributes of cortex based on Liley’s mean field model, and seek possible generation mechanisms of GPD rhythms from several factors. To these ends, the dynamical properties of simulated EEG consistent with neurophysiological features of human cortex are investigated, among which GPD patterns are our focus. Firstly, with different value sets of model parameters, we reproduce some typical simulation waveforms which are analogous to mammalian normal or abnormal brain activities detected by EEG. Or in other words, we put more emphasis on brain waveforms of GPD states, normal states, and low firing rate states in our numerical simulations, and mode transitions among different firing states are our main interests. Secondly, through analysis of maximum Lyapunov exponents and frequency spectrum, we give several mode transitions by varying synaptic connections between excitatory and inhibitory populations, which support the conjecture that selective changes of synaptic connections can trigger GPD states, such as in excitatory (AMPA receptors) and inhibitory neurotransmitters (GABA receptors). Thirdly, we stress the importance of time delay on neural population connections and find that they are free to transfer among different firing modes with appropriate time delays. Furthermore, considering the effects of external inputs to cerebral cortex, we verify that stimulation can lead to good controls on GPD patterns, including facilitation and elimination. Finally, we show that more dynamical rhythms can be produced when taking into account the cortico-cortical connections. These modeling results are expected to shed light into the pathophysiological mechanisms of GPD modes from a theoretical viewpoint.

In this paper, we investigated effects of deep brain stimulation (DBS) on Parkinson's disease (PD) when different target sites in the basal ganglia are stimulated. The targets which are investigated are subthalamic nucleus (STN), globus pallidus interna (GPi), and globus pallidus externa (GPe). For this purpose we used a computational model of the basal ganglia-thalamocortical system (BGTCS) with parameters calculated for mean field. This model is able to reproduce both the normal and Parkinsonian activities of basal ganglia, thalamus and cortex in a unified structure. In the present study, we used a mean-field model of the BGTCS, allowing a more complete framework to simulate DBS and to interpret its effects in the BGTCS. Our results suggest that DBS in the STN and GPe could restore the thalamus relay activity, while DBS in the GPi could inhibit it. Our results are compatible with the experimental and the clinical outcomes about the effects of DBS of different targets.

The following sections are included:

• Introduction

• Neutron Star Model

• Relativistic Mean Field Model with Kaons (Pions)

• Conclusions

• Bibliography