In this paper, we develop a new cortex-pallidum model to study the origin mechanism of Parkinson's oscillations in the cortex. In contrast to many previous models, the globus pallidus internal (GPi) and externa (GPe) both exert direct inhibitory feedback to the cortex. Using Hopf bifurcation analysis, two new critical conditions for oscillations, which can include the self-feedback projection of GPe, are obtained. In this paper, we find that the average discharge rate (ADR) is an important marker of oscillations, which can divide Hopf bifurcations into two types that can uniformly be used to explain the oscillation mechanism. Interestingly, the ADR of the cortex first increases and then decreases with increasing coupling weights that are projected to the GPe. Regarding the Hopf bifurcation critical conditions, the quantitative relationship between the inhibitory projection and excitatory projection to the GPe is monotonically increasing; in contrast, the relationship between different coupling weights in the cortex is monotonically decreasing. In general, the oscillation amplitude is the lowest near the bifurcation points and reaches the maximum value with the evolution of oscillations. The GPe is an effective target for deep brain stimulation to alleviate oscillations in the cortex.

The origin, location and cause of Parkinson's oscillation are not clear at present. In this paper, we establish a new cortex-basal ganglia model to study the origin mechanism of Parkinson beta oscillation. Unlike many previous models, this model includes two direct inhibitory projections from the globus pallidus external (GPe) segment to the cortex. We first obtain the critical calculation formula of Parkinson's oscillation by using the method of Quasilinear analysis. Different from previous studies, the formula obtained in this paper can include the self-feedback connection of GPe. Then, we use the bifurcation analysis method to systematically explain the influence of some key parameters on the oscillation. We find that the bifurcation principle of different cortical nuclei is different. In general, the increase of the discharge capacity of the nuclei will cause oscillation. In some special cases, the sharp reduction of the discharge rate of the nuclei will also cause oscillation. The direction of bifurcation simulation is consistent with the critical condition curve. Finally, we discuss the characteristics of oscillation amplitude. At the beginning of the oscillation, the amplitude is relatively small; with the evolution of oscillation, the amplitude will gradually strengthen. This is consistent with the experimental phenomenon. In most cases, the amplitude of cortical inhibitory nuclei (CIN) is greater than that of cortical excitatory nuclei (CEX), and the two direct inhibitory projections feedback from GPe can significantly reduce the amplitude gap between them. We calculate the main frequency of the oscillation generated in this model, which basically falls between 13 and 30 Hz, belonging to the typical beta frequency band oscillation. Some new results obtained in this paper can help to better understand the origin mechanism of Parkinson's disease and have guiding significance for the development of experiments.

In this paper, we introduce general distributed delays (in pathways between different neurons) and self-feedback distributed delays (in self-feedback pathways) into an improved cortex-basal ganglia (BG) network to study possible Hopf bifurcation mechanisms for Parkinson’s oscillations. Hopf bifurcation critical conditions for discrete delays are obtained, which compare well to numerical simulations. General distributed delays inhibit oscillations in BG, but self-feedback distributed delays promote oscillations. The different effects of strong and weak kernels on Hopf bifurcations mainly depend on self-feedback distributed delays, strong kernels are more conducive to the generation of oscillations when self-feedback distributed delays exceed a certain value. The excitatory feedback and inhibitory feedback from cortex to BG have different effects on oscillation. We define four different states (absolutely stable, conditional stable, conditional oscillation and absolutely oscillation) in this model, which can explain different mechanisms of oscillation origin. An increase in distributed delays can induce the transition between supercritical (SPH) and subcritical (SBH) Hopf bifurcations, which in turn may cause the transition of oscillations in different frequency bands, such as beta oscillations and alpha oscillations. Near the SBH, frequencies increase with an increase in discrete delays; and the trend is opposite at the SPH. In general, the amplitude of oscillation increases with the increase of firing activation level. Discrete delays can improve the firing activation level of the BG and suppress oscillations in its adjacent circuits. With the increase of distributed delays, oscillatory regions evolve regularly and the roles of general distributed delays and self-feedback distributed delays are contrary. Different types of delay have different effects on oscillation frequency. Discrete delay and self-feedback distributed delay have similar effects on oscillation frequency, but are different from general distributed delay. Whether the frequency will change with an increase in delay depends on coupling weights.

In this paper, we systematically study Parkinson oscillation origin mechanism, oscillation amplitude and frequency characteristics in an improved cortex-basal ganglia (EXN-INNSTN-GPe) resonance model. Firstly, Hopf bifurcation critical conditions T-1(0) and T-2(0) for heterogeneous delays are derived, which are greatly affected by coupling weights. The delay is a key factor to affect the frequency of oscillation, many frequency bands (such as gamma oscillation, beta oscillation, alpha oscillation, etc.) appear with an increase in delay. According to T-1(0) and T-2(0), the state of STN-GPe loop can be divided into four types: stable state, induced oscillation, spontaneous oscillation and compound oscillation, which represent different origin mechanism of oscillation. They have different amplitude characteristics, which increases with an increase in delay in most cases. The amplitude of the induced oscillation is smaller than the amplitude of the spontaneous oscillation in many cases. In the compound oscillation region, the EXN-STN antiphase oscillation mode can inhibit the oscillation amplitude in the STN-GPe. Induced oscillation may indicate that the oscillation origins in the cortex, which spreads to the STN-GPe loop with the evolution of disease. Self-feedback connections of EXN (W-EE) and INN (W-II) play a very important role in affecting the amplitude of induced oscillation. The induced oscillation amplitude of the GPe is increased with an increase in W-EE, but the trend is the opposite when it is affected by W-II. As an increase in coupling weight between the EXN and INN, the minimum of W-EE and W-II required to induce oscillation are both increased. The theoretical analysis is in good agreement with the numerical simulation, the mechanism obtained may inspire experimental and new theoretical research in Parkinson's disease. (C) 2022 Elsevier B.V. All rights reserved.

Different from many previous theoretical studies, this paper explores the regulatory mechanism of the spike and wave discharges (SWDs) in the reticular thalamic nucleus (TRN) by a dynamic computational model. We observe that the SWDs appears in the TRN by changing the coupling weights and delays in the thalamocortical circuit. The abundant poly-spikes wave discharges is also induced when the delay increases to large enough. These discharges can be inhibited by tuning the inhibitory output from the basal ganglia to the thalamus. The mechanisms of these waves can be explained in this model together with simulation results, which are different from the mechanisms in the cortex. The TRN is an important target in treating epilepsy, and the results may be a theoretical evidence for experimental study in the future.

BACKGROUND: The mechanism of prevention and treatment of epilepsy is a hot issue in theoretical research. OBJECTIVE: In this paper, we studied the control mechanism of the generalized spike-and-wave discharges (GSWD) by different types of external electrical stimulation acting on the subthalamic nucleus (STN) in a computational model. METHODS: Firstly, we analyzed the pathological mechanism of seizures, which were induced by different parameters in the thalamocortical (TC) circuit. Then, a voltage V was exerted in the STN. At last, we used the sine wave and square wave current stimulation in the STN. RESULTS: We found that seizures can be inhibited by tuning stimulus intensity into suitable range, and the direction of adjustment depended on the size of the parameter. We observed that the seizure can also be inhibited by tuning different parameters in current. CONCLUSIONS: Different inhibition mechanisms can be explained in this model, which may provide theoretical evidences for selecting the optimal treatment scheme in the clinical.

In this paper, we discuss the existence of solutions for a first-order nonlinear impulsive integro-differential equation with a general boundary value condition. New comparison principles are developed, and existence results for extremal solutions are obtained using the established principles and the monotone iterative technique. The results are more general than those of the periodic boundary problems, which may be widely applied in this field.

BACKGROUND:The selection of optimal target areas in the surgical treatment of epilepsy is always a difficult problem in medicine.OBJECTIVE:We employed a theoretical calculation model to explore the control mechanism of seizures by an external voltage stimulus acting in different nerve nuclei.METHODS:Theoretical analysis and numerical simulation were combined.RESULTS:The globus pallidus, excitatory pyramidal neurons, striatal D1 neurons, thalamic reticular nucleus and specific relay nuclei were selected, we analyzed that the electrical stimulation has different effects in these target areas.CONCLUSIONS:The data selected were reasonable in study, the results may give a theoretical support for similar studies in clinical.

In this paper, we study the parkinson oscillation mechanism in a computational model by bifurcation analysis and numerical simulation. Oscillatory activities can be induced by abnormal coupling weights and delays. The bidirectional Hopf bifurcation phenomena are found in simulations, which can uniformly explain the oscillation mechanism in this model. The Hopf1 represents the transition between the low firing rate stable state (SS) and oscillation state (OS), the Hopf2 represents the transition between the high firing rate stable state (HSS) and the OS, the mechanisms of them are different. The Hopf1 and Hopf2 bifurcations both show that when the state transfers from the stable region to the oscillation region, oscillatory activities originate from the beta frequency band or the gamma frequency band. We find that the changing trends of the frequency (DF) and oscillation amplitude (OSAM) are contrary in many cases. The effect of the delay in inhibitory pathways is greater than that of in excitatory pathways, and appropriate delays improve the discharge activation level (DAL) of the system. In all, we infer that oscillations can be induced by the follow factors: 1. Improvement of the DAL of the globus pallidus externa (GPe); 2. Reduce the DAL of the GPe from the HSS or the discharge saturation state; 3. The GPe can also resonate with the subthalamic nucleus (STN).

In this paper, we obtain solution sequences converging uniformly and quadratically to extremal solutions of an impulsive integro-differential system with delay. The main tools are the method of quasilinearization and the monotone iterative. The results obtained are more general and applicable than previous studies, especially the quadratic convergence of the solution for a class of integro-differential equations, which have been involved little by now.

In this paper, we study the generation and inhibition mechanisms of absence epilepsy appearing in the reticular thalamic nucleus (TRN) of the thalamus in a basal ganglia-corticothalamic network, which has been proven to be effective for simulating epileptic seizures in the brain. We found that the spike-and wave discharges (SWDs) activities (2-4 Hz) appeared in the TRN by adjusting the coupling weight in the excitatory pathway "SRN -> EPN" and the delay in the inhibitory pathway "TRN -> Thalamic relay nucleus (SRN)". The SWDs can disappear by controlling the firing state of the substantia nigra pars reticulata (SNr), which sends strong inhibitory projections to both the SRN and TRN in the model. Suppression of seizure activities may be realized through both the isolated pathway "SNr -> TRN" and the isolated pathway "SNr -> SRN", or through the competition mechanism between them. Unidirectional and bidirectional regulation phenomena were obtained in the simulations, and the mechanisms of these can be well explained in this model. The theoretical mechanism of epilepsy appearing in the TRN has received little attention to date, and we hope that the results shown in this paper can inspire new ideas for experimental study in the future. (c) 2020 Elsevier Ltd. All rights reserved.

In this paper, we establish some criteria for the stability of trivial solution of population growth models with impulsive perturbations. The working tools are based on the theory of generalized ordinary differential equations. Here, the conditions concerning the functions are more general than the classical ones.

BACKGROUND:Absence epilepsy (AE) is a systemic disease of the brain, which mainly occurs during childhood and adolescence. The control mechanism is still unclear, and few theoretical studies have been conducted to investigate this.OBJECTIVE:In this paper, we employed external direct voltage stimulation in the subthalamic nucleus to explore mechanisms that inhibit absence seizures.METHODS:All simulation results are obtained by the four-order Runge-Kutta method in the MATLAB environment. The inhibition mechanism can be inferred from the results.RESULTS:We found that the seizures may be inhibited by tuning the strength of the voltage to suitable ranges. This regulation may be achieved through the competition between the inhibitory projections from the basal ganglia to the thalamus.CONCLUSION:Because the mechanism underlying the treatment of epilepsy with a uniform direct current electric field is unclear, we hope that these results can inspire further experimental studies.

Excessive synchronous oscillation activities appear in the brain is a key pathological feature of Parkinson’s disease, the mechanism of which is still unclear. Although some previous studies indicated that $$\beta$$ oscillation (13–30 Hz) may directly originate in the network composed of the subthalamic nucleus (STN) and external globus pallidus (GPe) neurons, specific onset mechanisms of which are unclear, especially theoretical evidences in numerical simulation are still little. In this paper, we employ a STN–GPe mean-field model to explore the onset mechanism of Parkinson’s oscillation. In addition to $$\beta$$ oscillation, we find that some other common oscillation frequency bands can appear in this network, such as the $$\alpha$$ oscillation band (8–12 Hz), the $$\theta$$ oscillation band (4–7 Hz) and $$\delta$$ oscillation band (1–3 Hz). In addition to the coupling weight between the STN and GPe, the delay is also a critical factor to affect oscillatory activities, which can not be neglected compared to other parameters. Through simulation and analysis, we propose two possible theories may induce the system to transfer from the stable state to the oscillatory state in this model: (1). The oscillation activity can be induced when the firing activation level of the population increases to large enough; (2). In some special cases, the population may stay in the high firing rate stable state and the mean discharge rate of which is too large to induce oscillations, then oscillation activities may be induced as the MD decreases to moderate value. In most situations, the change trends of the MD and oscillation dominant frequency are contrary, which may be an important physiological phenomenon shown in this model. The delays and firing rates were obtained by simulating, which may be verified in the experiment in the future.

In this paper, a class of Nicholson’s blowflies model with patch structure and discontinuous harvesting terms is concerned. Some sufficient criteria are established for the existence, uniqueness and exponential convergence of positive pseudo almost periodic solutions of the model. The working tools are based on the contraction mapping principle, the theory of exponential dichotomy and Lyapunov functions. Moreover, an example is presented to illustrate the main findings.

Absence epileptiform activities are traditionally considered to be primarily induced by abnormal interactions between the cortical and thalamic neurons, which form the thalamocortical circuit in the brain. The basal ganglia, as an organizational unit in the brain, has close input and output relationships with the thalamocortical circuit. Although several studies report that the basal ganglia may participate in controlling and regulating absence epileptiform activities, to date, there have been no studies regarding whether the basal ganglia directly cause absence epileptiform activities. In this paper, we built a basal ganglia-corticothalamic network model to determine the role of basal ganglia in this disease. We determined that absence epileptiform activities might be directly induced by abnormal coupling strengths on certain pivotal pathways in the basal ganglia. These epileptiform activities can be well controlled by the coupling strengths of three major pathways that project from the thalamocortical network to the basal ganglia. The results implied that the substantia nigra pars compacta (SNc) can be considered to be the effective treatment target area for inhibiting epileptiform activities, which supports the observations of previous studies. Particularly, as a major contribution of this paper, we determined that the final presentation position of the epileptic slow spike waves is not limited to the cerebral cortex; these waves may additionally appear in the thalamus, striatal medium spiny neurons, striatal fast spiking interneuron, the SNc, subthalamic nucleus, substantia nigra pars reticulata and globus pallidus pars externa. In addition, consistent with several previous studies, the delay in the network was observed to be a critical factor for inducing transitions between different types of absence epileptiform activities. Our new model not only explains the onset and control mechanism but also provides a unified framework to study similar problems in neuron systems.

In this paper, we propose a new class of stochastic process called (?,?)-pseudo almost automorphic in p-mean, which generalize in a natural fashion the concept of square-mean almost automorphy and its various extensions. As application, we establish the existence, uniqueness of(?,?)-pseudo almost automorphic in p-distribution mild solution to nonautonomous stochastic functional integro-differential equations. Finally, an example is given to illustrate the significance of the main findings.

In this paper, we establish some sufficient criteria for the existence, uniqueness of discrete weighted pseudo asymptotically periodic mild solutions and asymptotic behavior for nonlinear fractional difference equations in Banach space, where the nonlinear perturbation is Lipschitz type, or non-Lipschitz type. The results are a consequence of application of different fixed point theorems, namely, the Banach contraction mapping principle, Leray-Schauder alternative theorem and Matkowski's fixed point technique.