Biophysically grounded mean-field models of neural populations under electrical stimulation

Entrainment by transcranial alternating current stimulation: Insights from models of cortical oscillations and dynamical systems theory

Signature of neuronal oscillations can be found in nearly every brain function. However, abnormal oscillatory activity is linked with several brain disorders. Transcranial alternating current stimulation (tACS) is a non-invasive brain stimulation technique that can potentially modulate neuronal oscillations and influence behavior both in health and disease. Yet, a complete understanding of how interacting networks of neurons are affected by tACS remains elusive. Entrainment effects by which tACS synchronizes neuronal oscillations is one of the main hypothesized mechanisms, as evidenced in animals and humans. Computational models of cortical oscillations may shed light on the entrainment effects of tACS, but current modeling studies lack specific guidelines to inform experimental investigations. This study addresses the existing gap in understanding the mechanisms of tACS effects on rhythmogenesis within the brain by providing a comprehensive overview of both theoretical and experimental perspectives. We explore the intricate interactions between oscillators and periodic stimulation through the lens of dynamical systems theory. Subsequently, we present a synthesis of experimental findings that demonstrate the effects of tACS on both individual neurons and collective oscillatory patterns in animal models and humans. Our review extends to computational investigations that elucidate the interplay between tACS and neuronal dynamics across diverse cortical network models. To illustrate these concepts, we conclude with a simple oscillatory neuron model, showcasing how fundamental theories of oscillatory behavior derived from dynamical systems, such as phase response of neurons to external perturbation, can account for the entrainment effects observed with tACS. Studies reviewed here render the necessity of integrated experimental and computational approaches for effective neuromodulation by tACS in health and disease.

Comparison of derivative-based and correlation-based methods to estimate effective connectivity in neural networks

Inferring and understanding the underlying connectivity structure of a system solely from the observed activity of its constituent components is a challenge in many areas of science. In neuroscience, techniques for estimating connectivity are paramount when attempting to understand the network structure of neural systems from their recorded activity patterns. To date, no universally accepted method exists for the inference of effective connectivity, which describes how the activity of a neural node mechanistically affects the activity of other nodes. Here, focussing on purely excitatory networks of small to intermediate size and continuous node dynamics, we provide a systematic comparison of different approaches for estimating effective connectivity. Starting with the Hopf neuron model in conjunction with known ground truth structural connectivity, we reconstruct the system's connectivity matrix using a variety of algorithms. We show that, in sparse non-linear networks with delays, combining a lagged-cross-correlation (LCC) approach with a recently published derivative-based covariance analysis method provides the most reliable estimation of the known ground truth connectivity matrix. We outline how the parameters of the Hopf model, including those controlling the bifurcation, noise, and delay distribution, affect this result. We also show that in linear networks, LCC has comparable performance to a method based on transfer entropy, at a drastically lower computational cost. We highlight that LCC works best for small sparse networks, and show how performance decreases in larger and less sparse networks. Applying the method to linear dynamics without time delays, we find that it does not outperform derivative-based methods. We comment on this finding in light of recent theoretical results for such systems. Employing the Hopf model, we then use the estimated structural connectivity matrix as the basis for a forward simulation of the system dynamics, in order to recreate the observed node activity patterns. We show that, under certain conditions, the best method, LCC, results in higher trace-to-trace correlations than derivative-based methods for sparse noise-driven systems. Finally, we apply the LCC method to empirical biological data. Choosing a suitable threshold for binarization, we reconstruct the structural connectivity of a subset of the nervous system of the nematode C. elegans. We show that the computationally simple LCC method performs better than another recently published, computationally more expensive reservoir computing-based method. We apply different methods to this dataset and find that they all lead to similar performances. Our results show that a comparatively simple method can be used to reliably estimate directed effective connectivity in sparse neural systems in the presence of spatio-temporal delays and noise. We provide concrete suggestions for the estimation of effective connectivity in a scenario common in biological research, where only neuronal activity of a small set of neurons, but not connectivity or single-neuron and synapse dynamics, are known.

A framework for optimal control of oscillations and synchrony applied to non-linear models of neural population dynamics

We adapt non-linear optimal control theory (OCT) to control oscillations and network synchrony and apply it to models of neural population dynamics. OCT is a mathematical framework to compute an efficient stimulation for dynamical systems. In its standard formulation, it requires a well-defined reference trajectory as target state. This requirement, however, may be overly restrictive for oscillatory targets, where the exact trajectory shape might not be relevant. To overcome this limitation, we introduce three alternative cost functionals to target oscillations and synchrony without specification of a reference trajectory. We successfully apply these cost functionals to single-node and network models of neural populations, in which each node is described by either the Wilson-Cowan model or a biophysically realistic high-dimensional mean-field model of exponential integrate-and-fire neurons. We compute efficient control strategies for four different control tasks. First, we drive oscillations from a stable stationary state at a particular frequency. Second, we switch between stationary and oscillatory stable states and find a translational invariance of the state-switching control signals. Third, we switch between in-phase and out-of-phase oscillations in a two-node network, where all cost functionals lead to identical OC signals in the minimum-energy limit. Finally, we (de-) synchronize an (a-) synchronously oscillating six-node network. In this setup, for the desynchronization task, we find very different control strategies for the three cost functionals. The suggested methods represent a toolbox that enables to include oscillatory phenomena into the framework of non-linear OCT without specification of an exact reference trajectory. However, task-specific adjustments of the optimization parameters have to be performed to obtain informative results.

Mathematical models on bone cell homeostasis and kinetics in the presence of electric fields: a review

The role of bioelectricity in regulating various physiological processes has attracted increasing scientific interest in implementing exogenous electrical stimulations as a therapeutic approach. In particular, electrical stimuli are used clinically in pre-/post-surgery patient care for the musculoskeletal tissues. The reported potential of electric fields (EF) to regulate bone cell homeostasis and kineticsin vitrohas further provoked more studies in this field of research. Various customised apparatuses have been developed, and a range of parameters for the applied EFs have been investigatedin vitrowith bone cells or mesenchymal stem cells. Additionally, biomaterials with conductive or piezo-electric properties have been designed to complement the enhancing effects of the EF on bone regeneration. Despite much research, there remained a significant gap in knowledge due to the diverse range of EF parameters available. Mathematical models are built to facilitate further understanding and zero in on an effective range of EF parametersin silico. However, the diverse range of EF parameters, experimental conditions, and reported analytical output of different works of literature were reported to possess significant variance, making it challenging to accurately model the fieldin silico. This review categorises the existing experimental approaches and the parameters used to distinguish the potential variables that apply to mathematical modelling. Furthermore, we will discuss existing modelling approaches and models available in the literature. With this, we will concisely highlight the need to categorise EF parameters, osteogenic differentiation initiators and research output.

A Mean Field to Capture Asynchronous Irregular Dynamics of Conductance-Based Networks of Adaptive Quadratic Integrate-and-Fire Neuron Models

Mean-field models are a class of models used in computational neuroscience to study the behavior of large populations of neurons. These models are based on the idea of representing the activity of a large number of neurons as the average behavior of mean-field variables. This abstraction allows the study of large-scale neural dynamics in a computationally efficient and mathematically tractable manner. One of these methods, based on a semianalytical approach, has previously been applied to different types of single-neuron models, but never to models based on a quadratic form. In this work, we adapted this method to quadratic integrate-and-fire neuron models with adaptation and conductance-based synaptic interactions. We validated the mean-field model by comparing it to the spiking network model. This mean-field model should be useful to model large-scale activity based on quadratic neurons interacting with conductance-based synapses.

intensity- and frequency-specific effects of transcranial alternating current stimulation are explained by network dynamics

Objective. Transcranial alternating current stimulation (tACS) can be used to non-invasively entrain neural activity and thereby cause changes in local neural oscillatory power. Despite its increased use in cognitive and clinical neuroscience, the fundamental mechanisms of tACS are still not fully understood.Approach. We developed a computational neuronal network model of two-compartment pyramidal neurons (PY) and inhibitory interneurons, which mimic the local cortical circuits. We modeled tACS with electric field strengths that are achievable in human applications. We then simulated intrinsic network activity and measured neural entrainment to investigate how tACS modulates ongoing endogenous oscillations.Main results. The intensity-specific effects of tACS are non-linear. At low intensities (<0.3 mV mm-1), tACS desynchronizes neural firing relative to the endogenous oscillations. At higher intensities (>0.3 mV mm-1), neurons are entrained to the exogenous electric field. We then further explore the stimulation parameter space and find that the entrainment of ongoing cortical oscillations also depends on stimulation frequency by following an Arnold tongue. Moreover, neuronal networks can amplify the tACS-induced entrainment via synaptic coupling and network effects. Our model shows that PY are directly entrained by the exogenous electric field and drive the inhibitory neurons.Significance. The results presented in this study provide a mechanistic framework for understanding the intensity- and frequency-specific effects of oscillating electric fields on neuronal networks. This is crucial for rational parameter selection for tACS in cognitive studies and clinical applications.

Transcranial alternating current stimulation does not affect microscale learning

A theory has been posited that microscale learning, which involves short intervals of a few seconds during explicit motor skill learning, considerably enhances performance. This phenomenon correlates with diminished beta-band activity in the frontal and parietal regions. However, there is a lack of neurophysiological studies regarding the relationship between microscale learning and implicit motor skill learning. In the present study, we aimed to determine the effects of transcranial alternating current stimulation (tACS) during short rest periods on microscale learning in an implicit motor task. We investigated the effects of 20-Hz β-tACS delivered during short rest periods while participants performed an implicit motor task. In Experiments 1 and 2, β-tACS targeted the right dorsolateral prefrontal cortex and the right frontoparietal network, respectively. The participants performed a finger-tapping task using their nondominant left hand, and microscale learning was separately analyzed for micro-online gains (MOnGs) and micro-offline gains (MOffGs). Contrary to our expectations, β-tACS exhibited no statistically significant effects on MOnGs or MOffGs in either Experiment 1 or Experiment 2. In addition, microscale learning during the performance of the implicit motor task was improved by MOffGs in the early learning phase and by MOnGs in the late learning phase. These results revealed that the stimulation protocol employed in this study did not affect microscale learning, indicating a novel aspect of microscale learning in implicit motor tasks. This is the first study to examine microscale learning in implicit motor tasks and may provide baseline information that will be useful in future studies.

Exploring global and local processes underlying alterations in resting-state functional connectivity and dynamics in schizophrenia

Introduction

We examined changes in large-scale functional connectivity and temporal dynamics and their underlying mechanisms in schizophrenia (ScZ) through measurements of resting-state functional magnetic resonance imaging (rs-fMRI) data and computational modelling.

Methods

The rs-fMRI measurements from patients with chronic ScZ (n=38) and matched healthy controls (n=43), were obtained through the public schizConnect repository. Computational models were constructed based on diffusion-weighted MRI scans and fit to the experimental rs-fMRI data.

Results

We found decreased large-scale functional connectivity across sensory and association areas and for all functional subnetworks for the ScZ group. Additionally global synchrony was reduced in patients while metastability was unaltered. Perturbations of the computational model revealed that decreased global coupling and increased background noise levels both explained the experimentally found deficits better than local changes to the GABAergic or glutamatergic system.

Discussion

The current study suggests that large-scale alterations in ScZ are more likely the result of global rather than local network changes.

Towards Optimization of Oscillatory Stimulation During Sleep

BACKGROUND

Oscillatory rhythms during sleep, such as slow oscillations (SOs) and spindles and, most importantly, their coupling, are thought to underlie processes of memory consolidation. External slow oscillatory transcranial direct current stimulation (so-tDCS) with a frequency of 0.75 Hz has been shown to improve this coupling and memory consolidation; however, effects varied quite markedly between individuals, studies, and species. In this study, we aimed to determine how precisely the frequency of stimulation must match the naturally occurring SO frequency in individuals to best improve SO-spindle coupling. Moreover, we systematically tested stimulation durations necessary to induce changes.

MATERIALS AND METHODS

We addressed these questions by comparing so-tDCS with individualized frequency to standardized frequency of 0.75 Hz in a within-subject design with 28 older participants during napping while stimulation train durations were systematically varied between 30 seconds, 2 minutes, and 5 minutes.

RESULTS

Stimulation trains as short as 30 seconds were sufficient to modulate the coupling between SOs and spindle activity. Contrary to our expectations, so-tDCS with standardized frequency indicated stronger aftereffects regarding SO-spindle coupling than individualized frequency. Angle and variance of spindle maxima occurrence during the SO cycle were similarly modulated.

CONCLUSIONS

In sum, short stimulation trains were sufficient to induce significant changes in sleep physiology, allowing for more trains of stimulation, which provides methodological advantages and possibly even larger behavioral effects in future studies. Regarding individualized stimulation frequency, further options of optimization need to be investigated, such as closed-loop stimulation, to calibrate stimulation frequency to the SO frequency at the time of stimulation onset.

CLINICAL TRIAL REGISTRATION

The Clinicaltrials.gov registration number for the study is NCT04714879.

Intensity- and frequency-specific effects of transcranial alternating current stimulation are explained by network dynamics

Transcranial alternating current stimulation (tACS) can be used to non-invasively entrain neural activity, and thereby cause changes in local neural oscillatory power. Despite an increased use in cognitive and clinical neuroscience, the fundamental mechanisms of tACS are still not fully understood. Here, we develop a computational neuronal network model of two-compartment pyramidal neurons and inhibitory interneurons which mimic the local cortical circuits. We model tACS with electric field strengths that are achievable in human applications. We then simulate intrinsic network activity and measure neural entrainment to investigate how tACS modulates ongoing endogenous oscillations. First, we show that intensity-specific effects of tACS are non-linear. At low intensities (<0.3 mV/mm), tACS desynchronizes neural firing relative to the endogenous oscillations. At higher intensities (>0.3 mV/mm), neurons are entrained to the exogenous electric field. We then further explore the stimulation parameter space and find that entrainment of ongoing cortical oscillations also depends on frequency by following an Arnold tongue. Moreover, neuronal networks can amplify the tACS induced entrainment via excitation-inhibition balance. Our model shows that pyramidal neurons are directly entrained by the exogenous electric field and drive the inhibitory neurons. Our findings can thus provide a mechanistic framework for understanding the intensity- and frequency- specific effects of oscillating electric fields on neuronal networks. This is crucial for rational parameters selection for tACS in cognitive studies and clinical applications.

Moving from phenomenological to predictive modelling: Progress and pitfalls of modelling brain stimulation in-silico

Brain stimulation is an increasingly popular neuromodulatory tool used in both clinical and research settings; however, the effects of brain stimulation, particularly those of non-invasive stimulation, are variable. This variability can be partially explained by an incomplete mechanistic understanding, coupled with a combinatorial explosion of possible stimulation parameters. Computational models constitute a useful tool to explore the vast sea of stimulation parameters and characterise their effects on brain activity. Yet the utility of modelling stimulation in-silico relies on its biophysical relevance, which needs to account for the dynamics of large and diverse neural populations and how underlying networks shape those collective dynamics. The large number of parameters to consider when constructing a model is no less than those needed to consider when planning empirical studies. This piece is centred on the application of phenomenological and biophysical models in non-invasive brain stimulation. We first introduce common forms of brain stimulation and computational models, and provide typical construction choices made when building phenomenological and biophysical models. Through the lens of four case studies, we provide an account of the questions these models can address, commonalities, and limitations across studies. We conclude by proposing future directions to fully realise the potential of computational models of brain stimulation for the design of personalized, efficient, and effective stimulation strategies.

Neurocognitive, physiological, and biophysical effects of transcranial alternating current stimulation

Transcranial alternating current stimulation (tACS) can modulate human neural activity and behavior. Accordingly, tACS has vast potential for cognitive research and brain disorder therapies. The stimulation generates oscillating electric fields in the brain that can bias neural spike timing, causing changes in local neural oscillatory power and cross-frequency and cross-area coherence. tACS affects cognitive performance by modulating underlying single or nested brain rhythms, local or distal synchronization, and metabolic activity. Clinically, stimulation tailored to abnormal neural oscillations shows promising results in alleviating psychiatric and neurological symptoms. We summarize the findings of tACS mechanisms, its use for cognitive applications, and novel developments for personalized stimulation.

Interplay between biochemical processes and network properties generates neuronal up and down states at the tripartite synapse

Neuronal up and down states have long been known to exist both in vitro and in vivo. A variety of functions and mechanisms have been proposed for their generation, but there has not been a clear connection between the functions and mechanisms. We explore the potential contribution of cellular-level biochemistry to the network-level mechanisms thought to underlie the generation of up and down states. We develop a neurochemical model of a single tripartite synapse, assumed to be within a network of similar tripartite synapses, to investigate possible function-mechanism links for the appearance of up and down states. We characterize the behavior of our model in different regions of parameter space and show that resource limitation at the tripartite synapse affects its ability to faithfully transmit input signals, leading to extinction-down states. Recovery of resources allows for "reignition" into up states. The tripartite synapse exhibits distinctive "regimes" of operation depending on whether ATP, neurotransmitter (glutamate), both, or neither, is limiting. Our model qualitatively matches the behavior of six disparate experimental systems, including both in vitro and in vivo models, without changing any model parameters except those related to the experimental conditions. We also explore the effects of varying different critical parameters within the model. Here we show that availability of energy, represented by ATP, and glutamate for neurotransmission at the cellular level are intimately related, and are capable of promoting state transitions at the network level as ignition and extinction phenomena. Our model is complementary to existing models of neuronal up and down states in that it focuses on cellular-level dynamics while still retaining essential network-level processes. Our model predicts the existence of a "final common pathway" of behavior at the tripartite synapse arising from scarcity of resources and may explain use dependence in the phenomenon of "local sleep." Ultimately, sleeplike behavior may be a fundamental property of networks of tripartite synapses.

Macroscopic dynamics of neural networks with heterogeneous spiking thresholds

Mean-field theory links the physiological properties of individual neurons to the emergent dynamics of neural population activity. These models provide an essential tool for studying brain function at different scales; however, for their application to neural populations on large scale, they need to account for differences between distinct neuron types. The Izhikevich single neuron model can account for a broad range of different neuron types and spiking patterns, thus rendering it an optimal candidate for a mean-field theoretic treatment of brain dynamics in heterogeneous networks. Here we derive the mean-field equations for networks of all-to-all coupled Izhikevich neurons with heterogeneous spiking thresholds. Using methods from bifurcation theory, we examine the conditions under which the mean-field theory accurately predicts the dynamics of the Izhikevich neuron network. To this end, we focus on three important features of the Izhikevich model that are subject here to simplifying assumptions: (i) spike-frequency adaptation, (ii) the spike reset conditions, and (iii) the distribution of single-cell spike thresholds across neurons. Our results indicate that, while the mean-field model is not an exact model of the Izhikevich network dynamics, it faithfully captures its different dynamic regimes and phase transitions. We thus present a mean-field model that can represent different neuron types and spiking dynamics. The model comprises biophysical state variables and parameters, incorporates realistic spike resetting conditions, and accounts for heterogeneity in neural spiking thresholds. These features allow for a broad applicability of the model as well as for a direct comparison to experimental data.

Computational models of neurotransmission at cerebellar synapses unveil the impact on network computation

The neuroscientific field benefits from the conjoint evolution of experimental and computational techniques, allowing for the reconstruction and simulation of complex models of neurons and synapses. Chemical synapses are characterized by presynaptic vesicle cycling, neurotransmitter diffusion, and postsynaptic receptor activation, which eventually lead to postsynaptic currents and subsequent membrane potential changes. These mechanisms have been accurately modeled for different synapses and receptor types (AMPA, NMDA, and GABA) of the cerebellar cortical network, allowing simulation of their impact on computation. Of special relevance is short-term synaptic plasticity, which generates spatiotemporal filtering in local microcircuits and controls burst transmission and information flow through the network. Here, we present how data-driven computational models recapitulate the properties of neurotransmission at cerebellar synapses. The simulation of microcircuit models is starting to reveal how diverse synaptic mechanisms shape the spatiotemporal profiles of circuit activity and computation.

Nonlinear optimal control of a mean-field model of neural population dynamics

We apply the framework of nonlinear optimal control to a biophysically realistic neural mass model, which consists of two mutually coupled populations of deterministic excitatory and inhibitory neurons. External control signals are realized by time-dependent inputs to both populations. Optimality is defined by two alternative cost functions that trade the deviation of the controlled variable from its target value against the “strength” of the control, which is quantified by the integrated 1- and 2-norms of the control signal. We focus on a bistable region in state space where one low- (“down state”) and one high-activity (“up state”) stable fixed points coexist. With methods of nonlinear optimal control, we search for the most cost-efficient control function to switch between both activity states. For a broad range of parameters, we find that cost-efficient control strategies consist of a pulse of finite duration to push the state variables only minimally into the basin of attraction of the target state. This strategy only breaks down once we impose time constraints that force the system to switch on a time scale comparable to the duration of the control pulse. Penalizing control strength via the integrated 1-norm (2-norm) yields control inputs targeting one or both populations. However, whether control inputs to the excitatory or the inhibitory population dominate, depends on the location in state space relative to the bifurcation lines. Our study highlights the applicability of nonlinear optimal control to understand neuronal processing under constraints better.

Exact mean-field models for spiking neural networks with adaptation

Networks of spiking neurons with adaption have been shown to be able to reproduce a wide range of neural activities, including the emergent population bursting and spike synchrony that underpin brain disorders and normal function. Exact mean-field models derived from spiking neural networks are extremely valuable, as such models can be used to determine how individual neurons and the network they reside within interact to produce macroscopic network behaviours. In the paper, we derive and analyze a set of exact mean-field equations for the neural network with spike frequency adaptation. Specifically, our model is a network of Izhikevich neurons, where each neuron is modeled by a two dimensional system consisting of a quadratic integrate and fire equation plus an equation which implements spike frequency adaptation. Previous work deriving a mean-field model for this type of network, relied on the assumption of sufficiently slow dynamics of the adaptation variable. However, this approximation did not succeed in establishing an exact correspondence between the macroscopic description and the realistic neural network, especially when the adaptation time constant was not large. The challenge lies in how to achieve a closed set of mean-field equations with the inclusion of the mean-field dynamics of the adaptation variable. We address this problem by using a Lorentzian ansatz combined with the moment closure approach to arrive at a mean-field system in the thermodynamic limit. The resulting macroscopic description is capable of qualitatively and quantitatively describing the collective dynamics of the neural network, including transition between states where the individual neurons exhibit asynchronous tonic firing and synchronous bursting. We extend the approach to a network of two populations of neurons and discuss the accuracy and efficacy of our mean-field approximations by examining all assumptions that are imposed during the derivation. Numerical bifurcation analysis of our mean-field models reveals bifurcations not previously observed in the models, including a novel mechanism for emergence of bursting in the network. We anticipate our results will provide a tractable and reliable tool to investigate the underlying mechanism of brain function and dysfunction from the perspective of computational neuroscience.

Automated Tuning of Closed-loop Neuromodulation Control Systems using Bayesian Optimization

Tuning the parameters of controllers to attain the best performance is a challenging task in designing effective closed-loop neuromodulation systems. In this paper, we present a distributed architecture for automated tuning and adaptation of closed-loop neuromodulation control systems. We use this approach for the automated parameter tuning of a Proportional-Integral (PI) neuromodulation controller using Bayesian optimization. We use a biophysically-grounded mean-field model of neural populations under electrical stimulation as a simulation environment for testing and prototyping the proposed framework and characterizing its performance. Our results demonstrate the feasibility of using Bayesian optimization for performance-based automated tuning of a PI controller in closed-loop set-point neuromodulation control tasks.

Alternating current stimulation entrains and connects cortical regions in a neural mass model

Transcranial alternating current stimulation (tACS) is a neuromodulatory technique that is widely used to investigate the functions of oscillations in the brain. Despite increasing usage in both research and clinical settings, the mechanisms of tACS are still not completely understood. To shed light on these mechanisms, we injected alternating current into a Jansen and Rit neural mass model. Two cortical columns were linked with long-range connections to examine how alternating current impacted cortical connectivity. Alternating current injected to both columns increased power and coherence at the stimulation frequency; however this effect was greatest at the model's resonant frequency. Varying the phase of stimulation impacted the time it took for entrainment to stabilize, an effect we believe is due to constructive and destructive inteference with endogenous membrane currents. The power output the model also depended on the phase of the stimulation between cortical columns. These results provide insight on the mechanisms of neurostimulation, by demonstrating that tACS increases both power and coherence at a neural network's resonant frequency, in a phase-dependent manner.

Cross-Frequency Slow Oscillation–Spindle Coupling in a Biophysically Realistic Thalamocortical Neural Mass Model

Sleep manifests itself by the spontaneous emergence of characteristic oscillatory rhythms, which often time-lock and are implicated in memory formation. Here, we analyze a neural mass model of the thalamocortical loop in which the cortical node can generate slow oscillations (approximately 1 Hz) while its thalamic component can generate fast sleep spindles of σ-band activity (12–15 Hz). We study the dynamics for different coupling strengths between the thalamic and cortical nodes, for different conductance values of the thalamic node's potassium leak and hyperpolarization-activated cation-nonselective currents, and for different parameter regimes of the cortical node. The latter are listed as follows: (1) a low activity (DOWN) state with noise-induced, transient excursions into a high activity (UP) state, (2) an adaptation induced slow oscillation limit cycle with alternating UP and DOWN states, and (3) a high activity (UP) state with noise-induced, transient excursions into the low activity (DOWN) state. During UP states, thalamic spindling is abolished or reduced. During DOWN states, the thalamic node generates sleep spindles, which in turn can cause DOWN to UP transitions in the cortical node. Consequently, this leads to spindle-induced UP state transitions in parameter regime (1), thalamic spindles induced in some but not all DOWN states in regime (2), and thalamic spindles following UP to DOWN transitions in regime (3). The spindle-induced σ-band activity in the cortical node, however, is typically the strongest during the UP state, which follows a DOWN state “window of opportunity” for spindling. When the cortical node is parametrized in regime (3), the model well explains the interactions between slow oscillations and sleep spindles observed experimentally during Non-Rapid Eye Movement sleep. The model is computationally efficient and can be integrated into large-scale modeling frameworks to study spatial aspects like sleep wave propagation.

The effect of alterations of schizophrenia-associated genes on gamma band oscillations

Abnormalities in the synchronized oscillatory activity of neurons in general and, specifically in the gamma band, might play a crucial role in the pathophysiology of schizophrenia. While these changes in oscillatory activity have traditionally been linked to alterations at the synaptic level, we demonstrate here, using computational modeling, that common genetic variants of ion channels can contribute strongly to this effect. Our model of primary auditory cortex highlights multiple schizophrenia-associated genetic variants that reduce gamma power in an auditory steady-state response task. Furthermore, we show that combinations of several of these schizophrenia-associated variants can produce similar effects as the more traditionally considered synaptic changes. Overall, our study provides a mechanistic link between schizophrenia-associated common genetic variants, as identified by genome-wide association studies, and one of the most robust neurophysiological endophenotypes of schizophrenia.

Effects of transcranial alternating current stimulation on spiking activity in computational models of single neocortical neurons

Neural oscillations are a key mechanism for information transfer in brain circuits. Rhythmic fluctuations of local field potentials control spike timing through cyclic membrane de- and hyperpolarization. Transcranial alternating current stimulation (tACS) is a non-invasive neuromodulation method which can directly interact with brain oscillatory activity by imposing an oscillating electric field on neurons. Despite its increasing use, the basic mechanisms of tACS are still not fully understood. Here, we investigate in a computational study the effects of tACS on morphologically realistic neurons with ongoing spiking activity. We characterize the membrane polarization as a function of electric field strength and subsequent effects on spiking activity in a set of 25 neurons from different neocortical layers. We find that tACS does not affect the firing rate of investigated neurons for electric field strengths applicable to human studies. However, we find that the applied electric fields entrain the spiking activity of large pyramidal neurons and large basket neurons at < 1 mV/mm field strengths. Our model results are in line with recent experimental studies and can provide a mechanistic framework to understand the effects of oscillating electric fields on single neuron activity. They highlight the importance of neuron morphology and biophysics in responsiveness to electrical stimulation.

Spatiotemporal Patterns of Adaptation-Induced Slow Oscillations in a Whole-Brain Model of Slow-Wave Sleep

During slow-wave sleep, the brain is in a self-organized regime in which slow oscillations (SOs) between up- and down-states travel across the cortex. While an isolated piece of cortex can produce SOs, the brain-wide propagation of these oscillations are thought to be mediated by the long-range axonal connections. We address the mechanism of how SOs emerge and recruit large parts of the brain using a whole-brain model constructed from empirical connectivity data in which SOs are induced independently in each brain area by a local adaptation mechanism. Using an evolutionary optimization approach, good fits to human resting-state fMRI data and sleep EEG data are found at values of the adaptation strength close to a bifurcation where the model produces a balance between local and global SOs with realistic spatiotemporal statistics. Local oscillations are more frequent, last shorter, and have a lower amplitude. Global oscillations spread as waves of silence across the undirected brain graph, traveling from anterior to posterior regions. These traveling waves are caused by heterogeneities in the brain network in which the connection strengths between brain areas determine which areas transition to a down-state first, and thus initiate traveling waves across the cortex. Our results demonstrate the utility of whole-brain models for explaining the origin of large-scale cortical oscillations and how they are shaped by the connectome.

neurolib: A Simulation Framework for Whole-Brain Neural Mass Modeling

neurolib is a computational framework for whole-brain modeling written in Python. It provides a set of neural mass models that represent the average activity of a brain region on a mesoscopic scale. In a whole-brain network model, brain regions are connected with each other based on biologically informed structural connectivity, i.e., the connectome of the brain. neurolib can load structural and functional datasets, set up a whole-brain model, manage its parameters, simulate it, and organize its outputs for later analysis. The activity of each brain region can be converted into a simulated BOLD signal in order to calibrate the model against empirical data from functional magnetic resonance imaging (fMRI). Extensive model analysis is made possible using a parameter exploration module, which allows one to characterize a model’s behavior as a function of changing parameters. An optimization module is provided for fitting models to multimodal empirical data using evolutionary algorithms. neurolib is designed to be extendable and allows for easy implementation of custom neural mass models, offering a versatile platform for computational neuroscientists for prototyping models, managing large numerical experiments, studying the structure–function relationship of brain networks, and for performing in-silico optimization of whole-brain models.

Software-Defined Workflows for Distributed Interoperable Closed-Loop Neuromodulation Control Systems

Closed-loop neuromodulation control systems facilitate regulating abnormal physiological processes by recording neurophysiological activities and modifying those activities through feedback loops. Designing such systems requires interoperable service composition, consisting of cycles. Workflow frameworks enable standard modular architectures, offering reproducible automated pipelines. However, those frameworks limit their support to executions represented by directed acyclic graphs (DAGs). DAGs need a pre-defined start and end execution step with no cycles, thus preventing the researchers from using the standard workflow languages as-is for closed-loop workflows and pipelines. In this paper, we present NEXUS, a workflow orchestration framework for distributed analytics systems. NEXUS proposes a Software-Defined Workflows approach, inspired by Software-Defined Networking (SDN), which separates the data flows across the service instances from the control flows. NEXUS enables creating interoperable workflows with closed loops by defining the workflows in a logically centralized approach, from microservices representing each execution step. The centralized NEXUS orchestrator facilitates dynamically composing and managing scientific workflows from the services and existing workflows, with minimal restrictions. NEXUS represents complex workflows as directed hypergraphs (DHGs) rather than DAGs. We illustrate a seamless execution of neuromodulation control systems by supporting loops in a workflow as the use case of NEXUS. Our evaluations highlight the feasibility, flexibility, performance, and scalability of NEXUS in modeling and executing closed-loop workflows.

Applications of optimal nonlinear control to a whole-brain network of FitzHugh-Nagumo oscillators

We apply the framework of optimal nonlinear control to steer the dynamics of a whole-brain network of FitzHugh-Nagumo oscillators. Its nodes correspond to the cortical areas of an atlas-based segmentation of the human cerebral cortex, and the internode coupling strengths are derived from diffusion tensor imaging data of the connectome of the human brain. Nodes are coupled using an additive scheme without delays and are driven by background inputs with fixed mean and additive Gaussian noise. Optimal control inputs to nodes are determined by minimizing a cost functional that penalizes the deviations from a desired network dynamic, the control energy, and spatially nonsparse control inputs. Using the strength of the background input and the overall coupling strength as order parameters, the network's state-space decomposes into regions of low- and high-activity fixed points separated by a high-amplitude limit cycle, all of which qualitatively correspond to the states of an isolated network node. Along the borders, however, additional limit cycles, asynchronous states, and multistability can be observed. Optimal control is applied to several state-switching and network synchronization tasks, and the results are compared to controllability measures from linear control theory for the same connectome. We find that intuitions from the latter about the roles of nodes in steering the network dynamics, which are solely based on connectome features, do not generally carry over to nonlinear systems, as had been previously implied. Instead, the role of nodes under optimal nonlinear control critically depends on the specified task and the system's location in state space. Our results shed new light on the controllability of brain network states and may serve as an inspiration for the design of new paradigms for noninvasive brain stimulation.

Improving the Effect of Transcranial Alternating Current Stimulation (tACS): A Systematic Review

With the development of electrical stimulation technology, traditional transcranial alternating current stimulation (tACS) technology has been found to have the drawback of not targeting a specific area accurately. Studies have shown that optimizing the number and position of electrodes during electrical stimulation has a very good effect on enhancing brain stimulation accuracy. At present, an increasing number of laboratories have begun to optimize tACS. However, there has been no study summarizing the optimization methods of tACS. Determining whether different optimization methods are effective and the optimization approach could provide information that could guide future tACS research. We describe the results of recent research on tACS optimization and integrate the optimization approaches of tACS in recent research. Optimization approaches can be classified into two groups: high-definition electrical stimulation and interference modulation electrical stimulation. The optimization methods can be divided into five categories: high-definition tACS, phase-shifted tACS, amplitude-modulated tACS, the temporally interfering (TI) method, and the intersectional short pulse (ISP) method. Finally, we summarize the latest research on hardware useful for tACS improvement and outline future directions.

Conducting double-blind placebo-controlled clinical trials of transcranial alternating current stimulation (tACS)

Many psychiatric and neurological illnesses can be conceptualized as oscillopathies defined as pathological changes in brain network oscillations. We previously proposed the application of rational design for the development of non-invasive brain stimulation for the modulation and restoration of cortical oscillations as a network therapeutic. Here, we show how transcranial alternating current stimulation (tACS), which applies a weak sine-wave electric current to the scalp, may serve as a therapeutic platform for the treatment of CNS illnesses. Recently, an initial series of double-blind, placebo-controlled treatment trials of tACS have been published. Here, we first map out the conceptual underpinnings of such trials with focus on target identification, engagement, and validation. Then, we discuss practical aspects that need to be considered for successful trial execution, with particular regards to ensuring successful study blind. Finally, we briefly review the few published double-blind tACS trials and conclude with a proposed roadmap to move the field forward with the goal of moving from pilot trials to convincing efficacy studies of tACS.

Effects of direct current stimulation on synaptic plasticity in a single neuron

BACKGROUND

Transcranial direct current stimulation (DCS) has lasting effects that may be explained by a boost in synaptic long-term potentiation (LTP). We hypothesized that this boost is the result of a modulation of somatic spiking in the postsynaptic neuron, as opposed to indirect network effects. To test this directly we record somatic spiking in a postsynaptic neuron during LTP induction with concurrent DCS.

METHODS

We performed rodent in-vitro patch-clamp recordings at the soma of individual CA1 pyramidal neurons. LTP was induced with theta-burst stimulation (TBS) applied concurrently with DCS. To test the causal role of somatic polarization, we manipulated polarization via current injections. We also used a computational multi-compartment neuron model that captures the effect of electric fields on membrane polarization and activity-dependent synaptic plasticity.

RESULTS

TBS-induced LTP was enhanced when paired with anodal DCS as well as depolarizing current injections. In both cases, somatic spiking during the TBS was increased, suggesting that evoked somatic activity is the primary factor affecting LTP modulation. However, the boost of LTP with DCS was less than expected given the increase in spiking activity alone. In some cells, we also observed DCS-induced spiking, suggesting DCS also modulates LTP via induced network activity. The computational model reproduces these results and suggests that they are driven by both direct changes in postsynaptic spiking and indirect changes due to network activity.

CONCLUSION

DCS enhances synaptic plasticity by increasing postsynaptic somatic spiking, but we also find that an increase in network activity may boost but also limit this enhancement.

Modelling the effects of transcranial alternating current stimulation on the neural encoding of speech in noise

Transcranial alternating current stimulation (tACS) can non-invasively modulate neuronal activity in the cerebral cortex, in particular at the frequency of the applied stimulation. Such modulation can matter for speech processing, since the latter involves the tracking of slow amplitude fluctuations in speech by cortical activity. tACS with a current signal that follows the envelope of a speech stimulus has indeed been found to influence the cortical tracking and to modulate the comprehension of the speech in background noise. However, how exactly tACS influences the speech-related cortical activity, and how it causes the observed effects on speech comprehension, remains poorly understood. A computational model for cortical speech processing in a biophysically plausible spiking neural network has recently been proposed. Here we extended the model to investigate the effects of different types of stimulation waveforms, similar to those previously applied in experimental studies, on the processing of speech in noise. We assessed in particular how well speech could be decoded from the neural network activity when paired with the exogenous stimulation. We found that, in the absence of current stimulation, the speech-in-noise decoding accuracy was comparable to the comprehension of speech in background noise of human listeners. We further found that current stimulation could alter the speech decoding accuracy by a few percent, comparable to the effects of tACS on speech-in-noise comprehension. Our simulations further allowed us to identify the parameters for the stimulation waveforms that yielded the largest enhancement of speech-in-noise encoding. Our model thereby provides insight into the potential neural mechanisms by which weak alternating current stimulation may influence speech comprehension and allows to screen a large range of stimulation waveforms for their effect on speech processing.

Neurostimulation stabilizes spiking neural networks by disrupting seizure-like oscillatory transitions

An improved understanding of the mechanisms underlying neuromodulatory approaches to mitigate seizure onset is needed to identify clinical targets for the treatment of epilepsy. Using a Wilson–Cowan-motivated network of inhibitory and excitatory populations, we examined the role played by intrinsic and extrinsic stimuli on the network’s predisposition to sudden transitions into oscillatory dynamics, similar to the transition to the seizure state. Our joint computational and mathematical analyses revealed that such stimuli, be they noisy or periodic in nature, exert a stabilizing influence on network responses, disrupting the development of such oscillations. Based on a combination of numerical simulations and mean-field analyses, our results suggest that high variance and/or high frequency stimulation waveforms can prevent multi-stability, a mathematical harbinger of sudden changes in network dynamics. By tuning the neurons’ responses to input, stimuli stabilize network dynamics away from these transitions. Furthermore, our research shows that such stabilization of neural activity occurs through a selective recruitment of inhibitory cells, providing a theoretical undergird for the known key role these cells play in both the healthy and diseased brain. Taken together, these findings provide new vistas on neuromodulatory approaches to stabilize neural microcircuit activity.