On the Role of Arkypallidal and Prototypical Neurons for Phase Transitions in the External Pallidum.

The external pallidum (globus pallidus pars externa [GPe]) plays a central role for basal ganglia functions and dynamics and, consequently, has been included in most computational studies of the basal ganglia. These studies considered the GPe as a homogeneous neural population. However, experimental studies have shown that the GPe contains at least two distinct cell types (prototypical and arkypallidal cells). In this work, we provide in silico insight into how pallidal heterogeneity modulates dynamic regimes inside the GPe and how they affect the GPe response to oscillatory input. We derive a mean-field model of the GPe system from a microscopic spiking neural network of recurrently coupled prototypical and arkypallidal neurons. Using bifurcation analysis, we examine the influence of dopamine-dependent changes of intrapallidal connectivity on the GPe dynamics. We find that increased self-inhibition of prototypical cells can induce oscillations, whereas increased inhibition of prototypical cells by arkypallidal cells leads to the emergence of a bistable regime. Furthermore, we show that oscillatory input to the GPe, arriving from striatum, leads to characteristic patterns of cross-frequency coupling observed at the GPe. Based on these findings, we propose two different hypotheses of how dopamine depletion at the GPe may lead to phase-amplitude coupling between the parkinsonian beta rhythm and a GPe-intrinsic γ rhythm. Finally, we show that these findings generalize to realistic spiking neural networks of sparsely coupled Type I excitable GPe neurons.SIGNIFICANCE STATEMENT Our work provides (1) insight into the theoretical implications of a dichotomous globus pallidus pars externa (GPe) organization, and (2) an exact mean-field model that allows for future investigations of the relationship between GPe spiking activity and local field potential fluctuations. We identify the major phase transitions that the GPe can undergo when subject to static or periodic input and link these phase transitions to the emergence of synchronized oscillations and cross-frequency coupling in the basal ganglia. Because of the close links between our model and experimental findings on the structure and dynamics of prototypical and arkypallidal cells, our results can be used to guide both experimental and computational studies on the role of the GPe for basal ganglia dynamics in health and disease.

Ion dynamics at the energy-deprived tripartite synapse.

The anatomical and functional organization of neurons and astrocytes at 'tripartite synapses' is essential for reliable neurotransmission, which critically depends on ATP. In low energy conditions, synaptic transmission fails, accompanied by a breakdown of ion gradients, changes in membrane potentials and cell swelling. The resulting cellular damage and cell death are causal to the often devastating consequences of an ischemic stroke. The severity of ischemic damage depends on the age and the brain region in which a stroke occurs, but the reasons for this differential vulnerability are far from understood. In the present study, we address this question by developing a comprehensive biophysical model of a glutamatergic synapse to identify key determinants of synaptic failure during energy deprivation. Our model is based on fundamental biophysical principles, includes dynamics of the most relevant ions, i.e., Na+, K+, Ca2+, Cl- and glutamate, and is calibrated with experimental data. It confirms the critical role of the Na+/K+-ATPase in maintaining ion gradients, membrane potentials and cell volumes. Our simulations demonstrate that the system exhibits two stable states, one physiological and one pathological. During energy deprivation, the physiological state may disappear, forcing a transit to the pathological state, which can be reverted when blocking voltage-gated Na+ and K+ channels. Our model predicts that the transition to the pathological state is favoured if the extracellular space fraction is small. A reduction in the extracellular space volume fraction, as, e.g. observed with ageing, will thus promote the brain's susceptibility to ischemic damage. Our work provides new insights into the brain's ability to recover from energy deprivation, with translational relevance for diagnosis and treatment of ischemic strokes.

Bistability and Stabilization of Human Visual Perception under Ambiguous Stimulation.

We discuss a computational model that describes stabilization of percept choices under intermittent viewing of an ambiguous visual stimulus at long stimulus intervals. Let T_off and T_on be the time that the stimulus is off and on, respectively. The behavior was studied by direct numerical simulation in a grid of (T_off, T_on) values in a 2007 paper of Noest, van Ee, Nijs, and van Wezel. They found that both alternating and repetitive sequences of percepts can appear stably, sometimes even for the same values of T_off and T_on. Longer T_off, however, always leads to a situation where, after transients, only repetitive sequences of percepts exist. We incorporate T_off and T_on explicitly as bifurcation parameters of an extended mathematical model of the perceptual choices. We elucidate the bifurcations of periodic orbits responsible for switching between alternating and repetitive sequences. We show that the stability borders of the alternating and repeating sequences in the (T_off, T_on) -parameter plane consist of curves of limit point and period-doubling bifurcations of periodic orbits. The stability regions overlap, resulting in a wedge with bistability of both sequences. We conclude by comparing our modeling results with the experimental results obtained by Noest, van Ee, Nijs, and van Wezel.

Pathological responses to single-pulse electrical stimuli in epilepsy: The role of feedforward inhibition.

Delineation of epileptogenic cortex in focal epilepsy patients may profit from single-pulse electrical stimulation during intracranial EEG recordings. Single-pulse electrical stimulation evokes early and delayed responses. Early responses represent connectivity. Delayed responses are a biomarker for epileptogenic cortex, but up till now, the precise mechanism generating delayed responses remains elusive. We used a data-driven modelling approach to study early and delayed responses. We hypothesized that delayed responses represent indirect responses triggered by early response activity and investigated this for 11 patients. Using two coupled neural masses, we modelled early and delayed responses by combining simulations and bifurcation analysis. An important feature of the model is the inclusion of feedforward inhibitory connections. The waveform of early responses can be explained by feedforward inhibition. Delayed responses can be viewed as second-order responses in the early response network which appear when input to a neural mass falls below a threshold forcing it temporarily to a spiking state. The combination of the threshold with noisy background input explains the typical stochastic appearance of delayed responses. The intrinsic excitability of a neural mass and the strength of its input influence the probability at which delayed responses to occur. Our work gives a theoretical basis for the use of delayed responses as a biomarker for the epileptogenic zone, confirming earlier clinical observations. The combination of early responses revealing effective connectivity, and delayed responses showing intrinsic excitability, makes single-pulse electrical stimulation an interesting tool to obtain data for computational models of epilepsy surgery.

A Comparison of Evoked and Non-evoked Functional Networks.

The growing interest in brain networks to study the brain's function in cognition and diseases has produced an increase in methods to extract these networks. Typically, each method yields a different network. Therefore, one may ask what the resulting networks represent. To address this issue we consider electrocorticography (ECoG) data where we compare three methods. We derive networks from on-going ECoG data using two traditional methods: cross-correlation (CC) and Granger causality (GC). Next, connectivity is probed actively using single pulse electrical stimulation (SPES). We compare the overlap in connectivity between these three methods as well as their ability to reveal well-known anatomical connections in the language circuit. We find that strong connections in the CC network form more or less a subset of the SPES network. GC and SPES are related more weakly, although GC connections coincide more frequently with SPES connections compared to non-existing SPES connections. Connectivity between the two major hubs in the language circuit, Broca's and Wernicke's area, is only found in SPES networks. Our results are of interest for the use of patient-specific networks obtained from ECoG. In epilepsy research, such networks form the basis for methods that predict the effect of epilepsy surgery. For this application SPES networks are interesting as they disclose more physiological connections compared to CC and GC networks.

Evoked directional network characteristics of epileptogenic tissue derived from single pulse electrical stimulation.

We investigated effective networks constructed from single pulse electrical stimulation (SPES) in epilepsy patients who underwent intracranial electrocorticography. Using graph analysis, we compared network characteristics of tissue within and outside the epileptogenic area. In 21 patients with subdural electrode grids (1 cm interelectrode distance), we constructed a binary, directional network derived from SPES early responses (<100 ms). We calculated in-degree, out-degree, betweenness centrality, the percentage of bidirectional, receiving and activating connections, and the percentage of connections toward the (non-)epileptogenic tissue for each node in the network. We analyzed whether these network measures were significantly different in seizure onset zone (SOZ)-electrodes compared to non-SOZ electrodes, in resected area (RA)-electrodes compared to non-RA electrodes, and in seizure free compared to not seizure-free patients. Electrodes in the SOZ/RA showed significantly higher values for in-degree and out-degree, both at group level, and at patient level, and more so in seizure-free patients. These differences were not observed for betweenness centrality. There were also more bidirectional and fewer receiving connections in the SOZ/RA in seizure-free patients. It appears that the SOZ/RA is densely connected with itself, with only little input arriving from non-SOZ/non-RA electrodes. These results suggest that meso-scale effective network measures are different in epileptogenic compared to normal brain tissue. Local connections within the SOZ/RA are increased and the SOZ/RA is relatively isolated from the surrounding cortex. This offers the prospect of enhanced prediction of epilepsy-prone brain areas using SPES.

Computational modeling of Adelta-fiber-mediated nociceptive detection of electrocutaneous stimulation.

Sensitization is an example of malfunctioning of the nociceptive pathway in either the peripheral or central nervous system. Using quantitative sensory testing, one can only infer sensitization, but not determine the defective subsystem. The states of the subsystems may be characterized using computational modeling together with experimental data. Here, we develop a neurophysiologically plausible model replicating experimental observations from a psychophysical human subject study. We study the effects of single temporal stimulus parameters on detection thresholds corresponding to a 0.5 detection probability. To model peripheral activation and central processing, we adapt a stochastic drift-diffusion model and a probabilistic hazard model to our experimental setting without reaction times. We retain six lumped parameters in both models characterizing peripheral and central mechanisms. Both models have similar psychophysical functions, but the hazard model is computationally more efficient. The model-based effects of temporal stimulus parameters on detection thresholds are consistent with those from human subject data.

Travelling waves in a neural field model with refractoriness.

At one level of abstraction neural tissue can be regarded as a medium for turning local synaptic activity into output signals that propagate over large distances via axons to generate further synaptic activity that can cause reverberant activity in networks that possess a mixture of excitatory and inhibitory connections. This output is often taken to be a firing rate, and the mathematical form for the evolution equation of activity depends upon a spatial convolution of this rate with a fixed anatomical connectivity pattern. Such formulations often neglect the metabolic processes that would ultimately limit synaptic activity. Here we reinstate such a process, in the spirit of an original prescription by Wilson and Cowan (Biophys J 12:1-24, 1972), using a term that multiplies the usual spatial convolution with a moving time average of local activity over some refractory time-scale. This modulation can substantially affect network behaviour, and in particular give rise to periodic travelling waves in a purely excitatory network (with exponentially decaying anatomical connectivity), which in the absence of refractoriness would only support travelling fronts. We construct these solutions numerically as stationary periodic solutions in a co-moving frame (of both an equivalent delay differential model as well as the original delay integro-differential model). Continuation methods are used to obtain the dispersion curve for periodic travelling waves (speed as a function of period), and found to be reminiscent of those for spatially extended models of excitable tissue. A kinematic analysis (based on the dispersion curve) predicts the onset of wave instabilities, which are confirmed numerically.

Tracking of nociceptive thresholds using adaptive psychophysical methods.

Psychophysical thresholds reflect the state of the underlying nociceptive mechanisms. For example, noxious events can activate endogenous analgesic mechanisms that increase the nociceptive threshold. Therefore, tracking thresholds over time facilitates the investigation of the dynamics of these underlying mechanisms. Threshold tracking techniques should use efficient methods for stimulus selection and threshold estimation. This study compares, in simulation and in human psychophysical experiments, the performance of different combinations of adaptive stimulus selection procedures and threshold estimation methods. Monte Carlo simulations were first performed to compare the bias and precision of threshold estimates produced by three different stimulus selection procedures (simple staircase, random staircase, and minimum entropy procedure) and two estimation methods (logistic regression and Bayesian estimation). Logistic regression and Bayesian estimations resulted in similar precision only when the prior probability distributions (PDs) were chosen appropriately. The minimum entropy and simple staircase procedures achieved the highest precision, while the random staircase procedure was the least sensitive to different procedure-specific settings. Next, the simple staircase and random staircase procedures, in combination with logistic regression, were compared in a human subject study (n = 30). Electrocutaneous stimulation was used to track the nociceptive perception threshold before, during, and after a cold pressor task, which served as the conditioning stimulus. With both procedures, habituation was detected, as well as changes induced by the conditioning stimulus. However, the random staircase procedure achieved a higher precision. We recommend using the random staircase over the simple staircase procedure, in combination with logistic regression, for nonstationary threshold tracking experiments.

Weak coupling of neurons enables very high-frequency and ultra-fast oscillations through the interplay of synchronized phase shifts.

Recently, in the past decade, high-frequency oscillations (HFOs), very high-frequency oscillations (VHFOs), and ultra-fast oscillations (UFOs) were reported in epileptic patients with drug-resistant epilepsy. However, to this day, the physiological origin of these events has yet to be understood. Our study establishes a mathematical framework based on bifurcation theory for investigating the occurrence of VHFOs and UFOs in depth EEG signals of patients with focal epilepsy, focusing on the potential role of reduced connection strength between neurons in an epileptic focus. We demonstrate that synchronization of a weakly coupled network can generate very and ultra high-frequency signals detectable by nearby microelectrodes. In particular, we show that a bistability region enables the persistence of phase-shift synchronized clusters of neurons. This phenomenon is observed for different hippocampal neuron models, including Morris-Lecar, Destexhe-Paré, and an interneuron model. The mechanism seems to be robust for small coupling, and it also persists with random noise affecting the external current. Our findings suggest that weakened neuronal connections could contribute to the production of oscillations with frequencies above 1000 Hz, which could advance our understanding of epilepsy pathology and potentially improve treatment strategies. However, further exploration of various coupling types and complex network models is needed.

We have built a mathematical framework to examine how a reduced neuronal coupling within an epileptic focus could lead to very high-frequency (VHFOs) and ultra-fast oscillations (UFOs) in depth EEG signals. By analyzing weakly coupled neurons, we found a bistability synchronization region where in-phase and anti-phase synchrony persist. These dynamics can be detected as very high-frequency EEG signals. The principle of weak coupling aligns with the disturbances in neuronal connections often observed in epilepsy; moreover, VHFOs are important markers of epileptogenicity. Our findings point to the potential significance of weakened neuronal connections in producing VHFOs and UFOs related to focal epilepsy. This could enhance our understanding of brain disorders. We emphasize the need for further investigations of weakly coupled neurons.

An in silico and in vitro human neuronal network model reveals cellular mechanisms beyond NaV1.1 underlying Dravet syndrome.

Human induced pluripotent stem cell (hiPSC)-derived neuronal networks on multi-electrode arrays (MEAs) provide a unique phenotyping tool to study neurological disorders. However, it is difficult to infer cellular mechanisms underlying these phenotypes. Computational modeling can utilize the rich dataset generated by MEAs, and advance understanding of disease mechanisms. However, existing models lack biophysical detail, or validation and calibration to relevant experimental data. We developed a biophysical in silico model that accurately simulates healthy neuronal networks on MEAs. To demonstrate the potential of our model, we studied neuronal networks derived from a Dravet syndrome (DS) patient with a missense mutation in SCN1A, encoding sodium channel NaV1.1. Our in silico model revealed that sodium channel dysfunctions were insufficient to replicate the in vitro DS phenotype, and predicted decreased slow afterhyperpolarization and synaptic strengths. We verified these changes in DS patient-derived neurons, demonstrating the utility of our in silico model to predict disease mechanisms.

Glial Chloride Homeostasis Under Transient Ischemic Stress.

High water permeabilities permit rapid adjustments of glial volume upon changes in external and internal osmolarity, and pathologically altered intracellular chloride concentrations ([Cl-]int) and glial cell swelling are often assumed to represent early events in ischemia, infections, or traumatic brain injury. Experimental data for glial [Cl-]int are lacking for most brain regions, under normal as well as under pathological conditions. We measured [Cl-]int in hippocampal and neocortical astrocytes and in hippocampal radial glia-like (RGL) cells in acute murine brain slices using fluorescence lifetime imaging microscopy with the chloride-sensitive dye MQAE at room temperature. We observed substantial heterogeneity in baseline [Cl-]int, ranging from 14.0 ± 2.0 mM in neocortical astrocytes to 28.4 ± 3.0 mM in dentate gyrus astrocytes. Chloride accumulation by the Na+-K+-2Cl- cotransporter (NKCC1) and chloride outward transport (efflux) through K+-Cl- cotransporters (KCC1 and KCC3) or excitatory amino acid transporter (EAAT) anion channels control [Cl-]int to variable extent in distinct brain regions. In hippocampal astrocytes, blocking NKCC1 decreased [Cl-]int, whereas KCC or EAAT anion channel inhibition had little effect. In contrast, neocortical astrocytic or RGL [Cl-]int was very sensitive to block of chloride outward transport, but not to NKCC1 inhibition. Mathematical modeling demonstrated that higher numbers of NKCC1 and KCC transporters can account for lower [Cl-]int in neocortical than in hippocampal astrocytes. Energy depletion mimicking ischemia for up to 10 min did not result in pronounced changes in [Cl-]int in any of the tested glial cell types. However, [Cl-]int changes occurred under ischemic conditions after blocking selected anion transporters. We conclude that stimulated chloride accumulation and chloride efflux compensate for each other and prevent glial swelling under transient energy deprivation.

Frequency-selectivity of a thalamocortical relay neuron during Parkinson's disease and deep brain stimulation: a computational study.

In this computational study, we investigated (i) the functional importance of correlated basal ganglia (BG) activity associated with Parkinson's disease (PD) motor symptoms by analysing the effects of globus pallidus internum (GPi) bursting frequency and synchrony on a thalamocortical (TC) relay neuron, which received GABAergic projections from this nucleus; (ii) the effects of subthalamic nucleus (STN) deep brain stimulation (DBS) on the response of the TC relay neuron to synchronized GPi oscillations; and (iii) the functional basis of the inverse relationship that has been reported between DBS frequency and stimulus amplitude, required to alleviate PD motor symptoms [A. L. Benabid et al. (1991)Lancet, 337, 403-406]. The TC relay neuron selectively responded to and relayed synchronized GPi inputs bursting at a frequency located in the range 2-25 Hz. Input selectivity of the TC relay neuron is dictated by low-threshold calcium current dynamics and passive membrane properties of the neuron. STN-DBS prevented the TC relay neuron from relaying synchronized GPi oscillations to cortex. Our model indicates that DBS alters BG output and input selectivity of the TC relay neuron, providing an explanation for the clinically observed inverse relationship between DBS frequency and stimulus amplitude.

Estimation and Identifiability of Model Parameters in Human Nociceptive Processing Using Yes-No Detection Responses to Electrocutaneous Stimulation.

Healthy or pathological states of nociceptive subsystems determine different stimulus-response relations measured from quantitative sensory testing. In turn, stimulus-response measurements may be used to assess these states. In a recently developed computational model, six model parameters characterize activation of nerve endings and spinal neurons. However, both model nonlinearity and limited information in yes-no detection responses to electrocutaneous stimuli challenge to estimate model parameters. Here, we address the question whether and how one can overcome these difficulties for reliable parameter estimation. First, we fit the computational model to experimental stimulus-response pairs by maximizing the likelihood. To evaluate the balance between model fit and complexity, i.e., the number of model parameters, we evaluate the Bayesian Information Criterion. We find that the computational model is better than a conventional logistic model regarding the balance. Second, our theoretical analysis suggests to vary the pulse width among applied stimuli as a necessary condition to prevent structural non-identifiability. In addition, the numerically implemented profile likelihood approach reveals structural and practical non-identifiability. Our model-based approach with integration of psychophysical measurements can be useful for a reliable assessment of states of the nociceptive system.

Dependence of Nociceptive Detection Thresholds on Physiological Parameters and Capsaicin-Induced Neuroplasticity: A Computational Study.

Physiological properties of peripheral and central nociceptive subsystems can be altered over time due to medical interventions. The effective change for the whole nociceptive system can be reflected in changes of psychophysical characteristics, e.g., detection thresholds. However, it is challenging to separate contributions of distinct altered mechanisms with measurements of thresholds only. Here, we aim to understand how these alterations affect Aδ-fiber-mediated nociceptive detection of electrocutaneous stimuli. First, with a neurophysiology-based model, we study the effects of single-model parameters on detection thresholds. Second, we derive an expression of model parameters determining the functional relationship between detection thresholds and the interpulse interval for double-pulse stimuli. Third, in a case study with topical capsaicin treatment, we translate neuroplasticity into plausible changes of model parameters. Model simulations qualitatively agree with changes in experimental detection thresholds. The simulations with individual forms of neuroplasticity confirm that nerve degeneration is the dominant mechanism for capsaicin-induced increases in detection thresholds. In addition, our study suggests that capsaicin-induced central plasticity may last at least 1 month.

Exploiting pallidal plasticity for stimulation in Parkinson's disease.

OBJECTIVE: Continuous application of high-frequency deep brain stimulation (DBS) often effectively reduces motor symptoms of Parkinson's disease patients. While there is a growing need for more effective and less traumatic stimulation, the exact mechanism of DBS is still unknown. Here, we present a methodology to exploit the plasticity of GABAergic synapses inside the external globus pallidus (GPe) for the optimization of DBS. APPROACH: Assuming the existence of spike-timing-dependent plasticity (STDP) at GABAergic GPe-GPe synapses, we simulate neural activity in a network model of the subthalamic nucleus and GPe. In particular, we test different DBS protocols in our model and quantify their influence on neural synchrony. MAIN RESULTS: In an exemplary set of biologically plausible model parameters, we show that STDP in the GPe has a direct influence on neural activity and especially the stability of firing patterns. STDP stabilizes both uncorrelated firing in the healthy state and correlated firing in the parkinsonian state. Alternative stimulation protocols such as coordinated reset stimulation can clearly profit from the stabilizing effect of STDP. These results are widely independent of the STDP learning rule. SIGNIFICANCE: Once the model settings, e.g., connection architectures, have been described experimentally, our model can be adjusted and directly applied in the development of novel stimulation protocols. More efficient stimulation leads to both minimization of side effects and savings in battery power.

Modeling focal epileptic activity in the Wilson-cowan model with depolarization block.

UNLABELLED: Measurements of neuronal signals during human seizure activity and evoked epileptic activity in experimental models suggest that, in these pathological states, the individual nerve cells experience an activity driven depolarization block, i.e. they saturate. We examined the effect of such a saturation in the Wilson-Cowan formalism by adapting the nonlinear activation function; we substituted the commonly applied sigmoid for a Gaussian function. We discuss experimental recordings during a seizure that support this substitution. Next we perform a bifurcation analysis on the Wilson-Cowan model with a Gaussian activation function. The main effect is an additional stable equilibrium with high excitatory and low inhibitory activity. Analysis of coupled local networks then shows that such high activity can stay localized or spread. Specifically, in a spatial continuum we show a wavefront with inhibition leading followed by excitatory activity. We relate our model simulations to observations of spreading activity during seizures. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1186/s13408-015-0019-4) contains supplementary material 1.

Phase-locking of epileptic spikes to ongoing delta oscillations in non-convulsive status epilepticus.

The EEG of patients in non-convulsive status epilepticus (NCSE) often displays delta oscillations or generalized spike-wave discharges. In some patients, these delta oscillations coexist with intermittent epileptic spikes. In this study we verify the prediction of a computational model of the thalamo-cortical system that these spikes are phase-locked to the delta oscillations. We subsequently describe the physiological mechanism underlying this observation as suggested by the model. It is suggested that the spikes reflect inhibitory stochastic fluctuations in the input to thalamo-cortical relay neurons and phase-locking is a consequence of differential excitability of relay neurons over the delta cycle. Further analysis shows that the observed phase-locking can be regarded as a stochastic precursor of generalized spike-wave discharges. This study thus provides an explanation of intermittent spikes during delta oscillations in NCSE and might be generalized to other encephathologies in which delta activity can be observed.

Comparing epileptiform behavior of mesoscale detailed models and population models of neocortex.

Two models of the neocortex are developed to study normal and pathologic neuronal activity. One model contains a detailed description of a neocortical microcolumn represented by 656 neurons, including superficial and deep pyramidal cells, four types of inhibitory neurons, and realistic synaptic contacts. Simulations show that neurons of a given type exhibit similar, synchronized behavior in this detailed model. This observation is captured by a population model that describes the activity of large neuronal populations with two differential equations with two delays. Both models appear to have similar sensitivity to variations of total network excitation. Analysis of the population model reveals the presence of multistability, which was also observed in various simulations of the detailed model.