Cellularly-driven differences in network synchronization propensity are differentially modulated by firing frequency

A spiking computational model for striatal cholinergic interneurons

Cholinergic interneurons in the striatum, also known as tonically active interneurons or TANs, are thought to have a strong effect on corticostriatal plasticity and on striatal activity and outputs, which in turn play a critical role in modulating downstream basal ganglia activity and movement. Striatal TANs can exhibit a variety of firing patterns and responses to synaptic inputs; furthermore, they have been found to display various surges and pauses in activity associated with sensory cues and reward delivery in learning as well as with motor tic production. To help explain the factors that contribute to TAN activity patterns and to provide a resource for future studies, we present a novel conductance-based computational model of a striatal TAN. We show that this model produces the various characteristic firing patterns observed in recordings of TANs. With a single baseline tuning associated with tonic firing, the model also captures a wide range of TAN behaviors found in previous experiments involving a variety of manipulations. In addition to demonstrating these results, we explain how various ionic currents in the model contribute to them. Finally, we use this model to explore the contributions of the acetylcholine released by TANs to the production of surges and pauses in TAN activity in response to strong excitatory inputs. These results provide predictions for future experimental testing that may help with efforts to advance our understanding of the role of TANs in reinforcement learning and in motor disorders such as Tourette's syndrome.

Dynamical Mechanism Underlying Scale-Free Network Reorganization in Low Acetylcholine States Corresponding to Slow Wave Sleep

Sleep is indispensable for most animals' cognitive functions, and is hypothesized to be a major factor in memory consolidation. Although we do not fully understand the mechanisms of network reorganisation driving memory consolidation, available data suggests that sleep-associated neurochemical changes may be important for such processes. In particular, global acetylcholine levels change across the sleep/wake cycle, with high cholinergic tone during wake and REM sleep and low cholinergic tone during slow wave sleep. Furthermore, experimental perturbation of cholinergic tone has been shown to impact memory storage. Through in silico modeling of neuronal networks, we show how spiking dynamics change in highly heterogenous networks under varying levels of cholinergic tone, with neuronal networks under high cholinergic modulation firing asynchronously and at high frequencies, while those under low cholinergic modulation exhibit synchronous patterns of activity. We further examined the network's dynamics and its reorganization mediated via changing levels of acetylcholine within the context of different scale-free topologies, comparing network activity within the hub cells, a small group of neurons having high degree connectivity, and with the rest of the network. We show a dramatic, state-dependent change in information flow throughout the network, with highly active hub cells integrating information in a high-acetylcholine state, and transferring it to rest of the network in a low-acetylcholine state. This result is experimentally corroborated by frequency-dependent frequency changes observed in vivo experiments. Together, these findings provide insight into how new neurons are recruited into memory traces during sleep, a mechanism which may underlie system memory consolidation.

Activity-mediated accumulation of potassium induces a switch in firing pattern and neuronal excitability type

During normal neuronal activity, ionic concentration gradients across a neuron’s membrane are often assumed to be stable. Prolonged spiking activity, however, can reduce transmembrane gradients and affect voltage dynamics. Based on mathematical modeling, we investigated the impact of neuronal activity on ionic concentrations and, consequently, the dynamics of action potential generation. We find that intense spiking activity on the order of a second suffices to induce changes in ionic reversal potentials and to consistently induce a switch from a regular to an intermittent firing mode. This transition is caused by a qualitative alteration in the system’s voltage dynamics, mathematically corresponding to a co-dimension-two bifurcation from a saddle-node on invariant cycle (SNIC) to a homoclinic orbit bifurcation (HOM). Our electrophysiological recordings in mouse cortical pyramidal neurons confirm the changes in action potential dynamics predicted by the models: (i) activity-dependent increases in intracellular sodium concentration directly reduce action potential amplitudes, an effect typically attributed solely to sodium channel inactivation; (ii) extracellular potassium accumulation switches action potential generation from tonic firing to intermittently interrupted output. Thus, individual neurons may respond very differently to the same input stimuli, depending on their recent patterns of activity and/or the current brain-state.

Ionic concentrations in the brain are not constant. We show that during intense neuronal activity, they can change on the order of seconds and even switch neuronal spiking patterns under identical stimulation from a regular firing mode to an intermittently interrupted one. Triggered by an accumulation of extracellular potassium, such a transition is caused by a specific, qualitative change in of the neuronal voltage dynamics—a so-called bifurcation—which affects crucial features of action-potential generation and bears consequences for how information is encoded and how neurons behave together in the network. Also, changes in intracellular sodium can induce measurable effects, like a reduction of spike amplitude that occurs independently of the fast amplitude effects attributed to sodium channel inactivation. Taken together, our results demonstrate that a neuron can respond very differently to the same stimulus, depending on its previous activity or the current brain state. This finding may be particularly relevant when other regulatory mechanisms of ionic homeostasis are challenged, for example, during pathological states of glial impairment or oxygen deprivation. Finally, categorization of cortical neurons as intrinsically bursting or regular spiking may be biased by the ionic concentrations at the time of the observation, highlighting the non-static nature of neuronal dynamics.

M-current induced Bogdanov-Takens bifurcation and switching of neuron excitability class

In this work, we consider a general conductance-based neuron model with the inclusion of the acetycholine sensitive, M-current. We study bifurcations in the parameter space consisting of the applied current [Formula: see text], the maximal conductance of the M-current [Formula: see text] and the conductance of the leak current [Formula: see text]. We give precise conditions for the model that ensure the existence of a Bogdanov-Takens (BT) point and show that such a point can occur by varying [Formula: see text] and [Formula: see text]. We discuss the case when the BT point becomes a Bogdanov-Takens-cusp (BTC) point and show that such a point can occur in the three-dimensional parameter space. The results of the bifurcation analysis are applied to different neuronal models and are verified and supplemented by numerical bifurcation diagrams generated using the package MATCONT. We conclude that there is a transition in the neuronal excitability type organised by the BT point and the neuron switches from Class-I to Class-II as conductance of the M-current increases.

Dynamics of a neuronal pacemaker in the weakly electric fish Apteronotus

The precise timing of neuronal activity is critical for normal brain function. In weakly electric fish, the medullary pacemaker network (PN) sets the timing for an oscillating electric organ discharge (EOD) used for electric sensing. This network is the most precise biological oscillator known, with sub-microsecond variation in oscillator period. The PN consists of two principle sets of neurons, pacemaker and relay cells, that are connected by gap junctions and normally fire in synchrony, one-to-one with each EOD cycle. However, the degree of gap junctional connectivity between these cells appears insufficient to provide the population averaging required for the observed temporal precision of the EOD. This has led to the hypothesis that individual cells themselves fire with high precision, but little is known about the oscillatory dynamics of these pacemaker cells. As a first step towards testing this hypothesis, we have developed a biophysical model of a pacemaker neuron action potential based on experimental recordings. We validated the model by comparing the changes in oscillatory dynamics produced by different experimental manipulations. Our results suggest that this relatively simple model can capture a large range of channel dynamics exhibited by pacemaker cells, and will thus provide a basis for future work on network synchrony and precision.

Phasic cholinergic signaling promotes emergence of local gamma rhythms in excitatory-inhibitory networks

Recent experimental results have shown that the detection of cues in behavioral attention tasks relies on transient increases of acetylcholine (ACh) release in frontal cortex and cholinergically driven oscillatory activity in the gamma frequency band (Howe et al. Journal of Neuroscience, 2017, 37, 3215). The cue-induced gamma rhythmic activity requires stimulation of M1 muscarinic receptors. Using biophysical computational modeling, we show that a network of excitatory (E) and inhibitory (I) neurons that initially displays asynchronous firing can generate transient gamma oscillatory activity in response to simulated brief pulses of ACh. ACh effects are simulated as transient modulation of the conductance of an M-type K⁺ current which is blocked by activation of muscarinic receptors and has significant effects on neuronal excitability. The ACh-induced effects on the M current conductance, g_Ks , change network dynamics to promote the emergence of network gamma rhythmicity through a Pyramidal-Interneuronal Network Gamma mechanism. Depending on connectivity strengths between and among E and I cells, gamma activity decays with the simulated g_Ks transient modulation or is sustained in the network after the g_Ks transient has completely dissipated. We investigated the sensitivity of the emergent gamma activity to synaptic strengths, external noise and simulated levels of g_Ks modulation. To address recent experimental findings that cholinergic signaling is likely spatially focused and dynamic, we show that localized g_Ks modulation can induce transient changes of cellular excitability in local subnetworks, subsequently causing population-specific gamma oscillations. These results highlight dynamical mechanisms underlying localization of ACh-driven responses and suggest that spatially localized, cholinergically induced gamma may contribute to selectivity in the processing of competing external stimuli, as occurs in attentional tasks.

Acetylcholine Mediates Dynamic Switching Between Information Coding Schemes in Neuronal Networks

Rate coding and phase coding are the two major coding modes seen in the brain. For these two modes, network dynamics must either have a wide distribution of frequencies for rate coding, or a narrow one to achieve stability in phase dynamics for phase coding. Acetylcholine (ACh) is a potent regulator of neural excitability. Acting through the muscarinic receptor, ACh reduces the magnitude of the potassium M-current, a hyperpolarizing current that builds up as neurons fire. The M-current contributes to several excitability features of neurons, becoming a major player in facilitating the transition between Type 1 (integrator) and Type 2 (resonator) excitability. In this paper we argue that this transition enables a dynamic switch between rate coding and phase coding as levels of ACh release change. When a network is in a high ACh state variations in synaptic inputs will lead to a wider distribution of firing rates across the network and this distribution will reflect the network structure or pattern of external input to the network. When ACh is low, network frequencies become narrowly distributed and the structure of a network or pattern of external inputs will be represented through phase relationships between firing neurons. This work provides insights into how modulation of neuronal features influences network dynamics and information processing across brain states.

Data-Driven Modeling of Cholinergic Modulation of Neural Microcircuits: Bridging Neurons, Synapses and Network Activity

Neuromodulators, such as acetylcholine (ACh), control information processing in neural microcircuits by regulating neuronal and synaptic physiology. Computational models and simulations enable predictions on the potential role of ACh in reconfiguring network activity. As a prelude into investigating how the cellular and synaptic effects of ACh collectively influence emergent network dynamics, we developed a data-driven framework incorporating phenomenological models of the physiology of cholinergic modulation of neocortical cells and synapses. The first-draft models were integrated into a biologically detailed tissue model of neocortical microcircuitry to investigate the effects of levels of ACh on diverse neuron types and synapses, and consequently on emergent network activity. Preliminary simulations from the framework, which was not tuned to reproduce any specific ACh-induced network effects, not only corroborate the long-standing notion that ACh desynchronizes spontaneous network activity, but also predict that a dose-dependent activation of ACh gives rise to a spectrum of neocortical network activity. We show that low levels of ACh, such as during non-rapid eye movement (nREM) sleep, drive microcircuit activity into slow oscillations and network synchrony, whereas high ACh concentrations, such as during wakefulness and REM sleep, govern fast oscillations and network asynchrony. In addition, spontaneous network activity modulated by ACh levels shape spike-time cross-correlations across distinct neuronal populations in strikingly different ways. These effects are likely due to the regulation of neurons and synapses caused by increasing levels of ACh, which enhances cellular excitability and decreases the efficacy of local synaptic transmission. We conclude by discussing future directions to refine the biological accuracy of the framework, which will extend its utility and foster the development of hypotheses to investigate the role of neuromodulators in neural information processing.

Dynamical Mechanism of Hyperpolarization-Activated Non-specific Cation Current Induced Resonance and Spike-Timing Precision in a Neuronal Model

Hyperpolarization-activated cyclic nucleotide-gated cation current (I_h) plays important roles in the achievement of many physiological/pathological functions in the nervous system by modulating the electrophysiological activities, such as the rebound (spike) to hyperpolarization stimulations, subthreshold membrane resonance to sinusoidal currents, and spike-timing precision to stochastic factors. In the present paper, with increasing g_h (conductance of I_h), the rebound (spike) and subthreshold resonance appear and become stronger, and the variability of the interspike intervals (ISIs) becomes lower, i.e., the enhancement of spike-timing precision, which are simulated in a conductance-based theoretical model and well explained by the nonlinear concept of bifurcation. With increasing g_h, the stable node to stable focus, to coexistence behavior, and to firing via the codimension-1 bifurcations (Hopf bifurcation, saddle-node bifurcation, saddle-node bifurcations on an invariant circle, and saddle homoclinic orbit) and codimension-2 bifurcations such as Bogdanov-Takens (BT) point related to the transition between saddle-node and Hopf bifurcations, are acquired with 1- and 2-parameter bifurcation analysis. The decrease of variability of ISIs with increasing g_h is induced by the fast decrease of the standard deviation of ISIs, which is related to the increase of the capacity of resisting noisy disturbance due to the firing becomes far away from the bifurcation point. The enhancement of the rebound (spike) with increasing g_h builds up a relationship to the decrease of the capacity of resisting disturbance like the hyperpolarization stimulus as the resting state approaches the bifurcation point. The “typical”-resonance and non-resonance appear in the parameter region of the stable focus and node far away from the bifurcation points, respectively. The complex or “strange” dynamics, such as the “weak”-resonance for the stable node near the transition point between the stable node and focus and the non-resonance for the stable focus close to the codimension-1 and −2 bifurcation points, are discussed.

Dichotomous Dynamics in E-I Networks with Strongly and Weakly Intra-connected Inhibitory Neurons

The interconnectivity between excitatory and inhibitory neural networks informs mechanisms by which rhythmic bursts of excitatory activity can be produced in the brain. One such mechanism, Pyramidal Interneuron Network Gamma (PING), relies primarily upon reciprocal connectivity between the excitatory and inhibitory networks, while also including intra-connectivity of inhibitory cells. The causal relationship between excitatory activity and the subsequent burst of inhibitory activity is of paramount importance to the mechanism and has been well studied. However, the role of the intra-connectivity of the inhibitory network, while important for PING, has not been studied in detail, as most analyses of PING simply assume that inhibitory intra-connectivity is strong enough to suppress subsequent firing following the initial inhibitory burst. In this paper we investigate the role that the strength of inhibitory intra-connectivity plays in determining the dynamics of PING-style networks. We show that networks with weak inhibitory intra-connectivity exhibit variations in burst dynamics of both the excitatory and inhibitory cells that are not obtained with strong inhibitory intra-connectivity. Networks with weak inhibitory intra-connectivity exhibit excitatory rhythmic bursts with weak excitatory-to-inhibitory synapses for which classical PING networks would show no rhythmic activity. Additionally, variations in dynamics of these networks as the excitatory-to-inhibitory synaptic weight increases illustrates the important role that consistent pattern formation in the inhibitory cells serves in maintaining organized and periodic excitatory bursts. Finally, motivated by these results and the known diversity of interneurons, we show that a PING-style network with two inhibitory subnetworks, one strongly intra-connected and one weakly intra-connected, exhibits organized and periodic excitatory activity over a larger parameter regime than networks with a homogeneous inhibitory population. Taken together, these results serve to better articulate the role of inhibitory intra-connectivity in generating PING-like rhythms, while also revealing how heterogeneity amongst inhibitory synapses might make such rhythms more robust to a variety of network parameters.

Synaptic Impairment and Robustness of Excitatory Neuronal Networks with Different Topologies

Synaptic deficiencies are a known hallmark of neurodegenerative diseases, but the diagnosis of impaired synapses on the cellular level is not an easy task. Nonetheless, changes in the system-level dynamics of neuronal networks with damaged synapses can be detected using techniques that do not require high spatial resolution. This paper investigates how the structure/topology of neuronal networks influences their dynamics when they suffer from synaptic loss. We study different neuronal network structures/topologies by specifying their degree distributions. The modes of the degree distribution can be used to construct networks that consist of rich clubs and resemble small world networks, as well. We define two dynamical metrics to compare the activity of networks with different structures: persistent activity (namely, the self-sustained activity of the network upon removal of the initial stimulus) and quality of activity (namely, percentage of neurons that participate in the persistent activity of the network). Our results show that synaptic loss affects the persistent activity of networks with bimodal degree distributions less than it affects random networks. The robustness of neuronal networks enhances when the distance between the modes of the degree distribution increases, suggesting that the rich clubs of networks with distinct modes keep the whole network active. In addition, a tradeoff is observed between the quality of activity and the persistent activity. For a range of distributions, both of these dynamical metrics are considerably high for networks with bimodal degree distribution compared to random networks. We also propose three different scenarios of synaptic impairment, which may correspond to different pathological or biological conditions. Regardless of the network structure/topology, results demonstrate that synaptic loss has more severe effects on the activity of the network when impairments are correlated with the activity of the neurons.

Intrinsic Cellular Properties and Connectivity Density Determine Variable Clustering Patterns in Randomly Connected Inhibitory Neural Networks

The plethora of inhibitory interneurons in the hippocampus and cortex play a pivotal role in generating rhythmic activity by clustering and synchronizing cell firing. Results of our simulations demonstrate that both the intrinsic cellular properties of neurons and the degree of network connectivity affect the characteristics of clustered dynamics exhibited in randomly connected, heterogeneous inhibitory networks. We quantify intrinsic cellular properties by the neuron's current-frequency relation (IF curve) and Phase Response Curve (PRC), a measure of how perturbations given at various phases of a neurons firing cycle affect subsequent spike timing. We analyze network bursting properties of networks of neurons with Type I or Type II properties in both excitability and PRC profile; Type I PRCs strictly show phase advances and IF curves that exhibit frequencies arbitrarily close to zero at firing threshold while Type II PRCs display both phase advances and delays and IF curves that have a non-zero frequency at threshold. Type II neurons whose properties arise with or without an M-type adaptation current are considered. We analyze network dynamics under different levels of cellular heterogeneity and as intrinsic cellular firing frequency and the time scale of decay of synaptic inhibition are varied. Many of the dynamics exhibited by these networks diverge from the predictions of the interneuron network gamma (ING) mechanism, as well as from results in all-to-all connected networks. Our results show that randomly connected networks of Type I neurons synchronize into a single cluster of active neurons while networks of Type II neurons organize into two mutually exclusive clusters segregated by the cells' intrinsic firing frequencies. Networks of Type II neurons containing the adaptation current behave similarly to networks of either Type I or Type II neurons depending on network parameters; however, the adaptation current creates differences in the cluster dynamics compared to those in networks of Type I or Type II neurons. To understand these results, we compute neuronal PRCs calculated with a perturbation matching the profile of the synaptic current in our networks. Differences in profiles of these PRCs across the different neuron types reveal mechanisms underlying the divergent network dynamics.

Coding Properties of Three Intrinsically Distinct Retinal Ganglion Cells under Periodic Stimuli: A Computational Study

As the sole output neurons in the retina, ganglion cells play significant roles in transforming visual information into spike trains, and then transmitting them to the higher visual centers. However, coding strategies that retinal ganglion cells (RGCs) adopt to accomplish these processes are not completely clear yet. To clarify these issues, we investigate the coding properties of three types of RGCs (repetitive spiking, tonic firing, and phasic firing) by two different measures (spike-rate and spike-latency). Model results show that for periodic stimuli, repetitive spiking RGC and tonic RGC exhibit similar spike-rate patterns. Their spike- rates decrease gradually with increased stimulus frequency, moreover, variation of stimulus amplitude would change the two RGCs' spike-rate patterns. For phasic RGC, it activates strongly at medium levels of frequency when the stimulus amplitude is low. While if high stimulus amplitude is applied, phasic RGC switches to respond strongly at low frequencies. These results suggest that stimulus amplitude is a prominent factor in regulating RGCs in encoding periodic signals. Similar conclusions can be drawn when analyzes spike-latency patterns of the three RGCs. More importantly, the above phenomena can be accurately reproduced by Hodgkin's three classes of neurons, indicating that RGCs can perform the typical three classes of firing dynamics, depending on the distinctions of ion channel densities. Consequently, model results from the three RGCs may be not specific, but can also applicable to neurons in other brain regions which exhibit part(s) or all of the Hodgkin's three excitabilities.

Interplay between excitability type and distributions of neuronal connectivity determines neuronal network synchronization

While the interplay between neuronal excitability properties and global properties of network topology is known to affect network propensity for synchronization, it is not clear how detailed characteristics of these properties affect spatiotemporal pattern formation. Here we study mixed networks, composed of neurons having type I and/or type II phase response curves, with varying distributions of local and random connections and show that not only average network properties, but also the connectivity distribution statistics, significantly affect network synchrony. Namely, we study networks with fixed networkwide properties, but vary the number of random connections that nodes project. We show that varying node excitability (type I vs type II) influences network synchrony most dramatically for systems with long-tailed distributions of the number of random connections per node. This indicates that a cluster of even a few highly rewired cells with a high propensity for synchronization can alter the degree of synchrony in the network as a whole. We show this effect generally on a network of coupled Kuramoto oscillators and investigate the impact of this effect more thoroughly in pulse-coupled networks of biophysical neurons.

Network burst dynamics under heterogeneous cholinergic modulation of neural firing properties and heterogeneous synaptic connectivity

The characteristics of neural network activity depend on intrinsic neural properties and synaptic connectivity in the network. In brain networks, both of these properties are critically affected by the type and levels of neuromodulators present. The expression of many of the most powerful neuromodulators, including acetylcholine (ACh), varies tonically and phasically with behavioural state, leading to dynamic, heterogeneous changes in intrinsic neural properties and synaptic connectivity properties. Namely, ACh significantly alters neural firing properties as measured by the phase response curve in a manner that has been shown to alter the propensity for network synchronization. The aim of this simulation study was to build an understanding of how heterogeneity in cholinergic modulation of neural firing properties and heterogeneity in synaptic connectivity affect the initiation and maintenance of synchronous network bursting in excitatory networks. We show that cells that display different levels of ACh modulation have differential roles in generating network activity: weakly modulated cells are necessary for burst initiation and provide synchronizing drive to the rest of the network, whereas strongly modulated cells provide the overall activity level necessary to sustain burst firing. By applying several quantitative measures of network activity, we further show that the existence of network bursting and its characteristics, such as burst duration and intraburst synchrony, are dependent on the fraction of cell types providing the synaptic connections in the network. These results suggest mechanisms underlying ACh modulation of brain oscillations and the modulation of seizure activity during sleep states.

Formation and Dynamics of Waves in a Cortical Model of Cholinergic Modulation

Acetylcholine (ACh) is a regulator of neural excitability and one of the neurochemical substrates of sleep. Amongst the cellular effects induced by cholinergic modulation are a reduction in spike-frequency adaptation (SFA) and a shift in the phase response curve (PRC). We demonstrate in a biophysical model how changes in neural excitability and network structure interact to create three distinct functional regimes: localized asynchronous, traveling asynchronous, and traveling synchronous. Our results qualitatively match those observed experimentally. Cortical activity during slow wave sleep (SWS) differs from that during REM sleep or waking states. During SWS there are traveling patterns of activity in the cortex; in other states stationary patterns occur. Our model is a network composed of Hodgkin-Huxley type neurons with a M-current regulated by ACh. Regulation of ACh level can account for dynamical changes between functional regimes. Reduction of the magnitude of this current recreates the reduction in SFA the shift from a type 2 to a type 1 PRC observed in the presence of ACh. When SFA is minimal (in waking or REM sleep state, high ACh) patterns of activity are localized and easily pinned by network inhomogeneities. When SFA is present (decreasing ACh), traveling waves of activity naturally arise. A further decrease in ACh leads to a high degree of synchrony within traveling waves. We also show that the level of ACh determines how sensitive network activity is to synaptic heterogeneity. These regimes may have a profound functional significance as stationary patterns may play a role in the proper encoding of external input as memory and traveling waves could lead to synaptic regularization, giving unique insights into the role and significance of ACh in determining patterns of cortical activity and functional differences arising from the patterns.

Input-output relation and energy efficiency in the neuron with different spike threshold dynamics

Neuron encodes and transmits information through generating sequences of output spikes, which is a high energy-consuming process. The spike is initiated when membrane depolarization reaches a threshold voltage. In many neurons, threshold is dynamic and depends on the rate of membrane depolarization (dV/dt) preceding a spike. Identifying the metabolic energy involved in neural coding and their relationship to threshold dynamic is critical to understanding neuronal function and evolution. Here, we use a modified Morris-Lecar model to investigate neuronal input-output property and energy efficiency associated with different spike threshold dynamics. We find that the neurons with dynamic threshold sensitive to dV/dt generate discontinuous frequency-current curve and type II phase response curve (PRC) through Hopf bifurcation, and weak noise could prohibit spiking when bifurcation just occurs. The threshold that is insensitive to dV/dt, instead, results in a continuous frequency-current curve, a type I PRC and a saddle-node on invariant circle bifurcation, and simultaneously weak noise cannot inhibit spiking. It is also shown that the bifurcation, frequency-current curve and PRC type associated with different threshold dynamics arise from the distinct subthreshold interactions of membrane currents. Further, we observe that the energy consumption of the neuron is related to its firing characteristics. The depolarization of spike threshold improves neuronal energy efficiency by reducing the overlap of Na(+) and K(+) currents during an action potential. The high energy efficiency is achieved at more depolarized spike threshold and high stimulus current. These results provide a fundamental biophysical connection that links spike threshold dynamics, input-output relation, energetics and spike initiation, which could contribute to uncover neural encoding mechanism.

Synchronization properties of heterogeneous neuronal networks with mixed excitability type

We study the synchronization of neuronal networks with dynamical heterogeneity, showing that network structures with the same propensity for synchronization (as quantified by master stability function analysis) may develop dramatically different synchronization properties when heterogeneity is introduced with respect to neuronal excitability type. Specifically, we investigate networks composed of neurons with different types of phase response curves (PRCs), which characterize how oscillating neurons respond to excitatory perturbations. Neurons exhibiting type 1 PRC respond exclusively with phase advances, while neurons exhibiting type 2 PRC respond with either phase delays or phase advances, depending on when the perturbation occurs. We find that Watts-Strogatz small world networks transition to synchronization gradually as the proportion of type 2 neurons increases, whereas scale-free networks may transition gradually or rapidly, depending upon local correlations between node degree and excitability type. Random placement of type 2 neurons results in gradual transition to synchronization, whereas placement of type 2 neurons as hubs leads to a much more rapid transition, showing that type 2 hub cells easily "hijack" neuronal networks to synchronization. These results underscore the fact that the degree of synchronization observed in neuronal networks is determined by a complex interplay between network structure and the dynamical properties of individual neurons, indicating that efforts to recover structural connectivity from dynamical correlations must in general take both factors into account.

Interplay of intrinsic and synaptic conductances in the generation of high-frequency oscillations in interneuronal networks with irregular spiking

High-frequency oscillations (above 30 Hz) have been observed in sensory and higher-order brain areas, and are believed to constitute a general hallmark of functional neuronal activation. Fast inhibition in interneuronal networks has been suggested as a general mechanism for the generation of high-frequency oscillations. Certain classes of interneurons exhibit subthreshold oscillations, but the effect of this intrinsic neuronal property on the population rhythm is not completely understood. We study the influence of intrinsic damped subthreshold oscillations in the emergence of collective high-frequency oscillations, and elucidate the dynamical mechanisms that underlie this phenomenon. We simulate neuronal networks composed of either Integrate-and-Fire (IF) or Generalized Integrate-and-Fire (GIF) neurons. The IF model displays purely passive subthreshold dynamics, while the GIF model exhibits subthreshold damped oscillations. Individual neurons receive inhibitory synaptic currents mediated by spiking activity in their neighbors as well as noisy synaptic bombardment, and fire irregularly at a lower rate than population frequency. We identify three factors that affect the influence of single-neuron properties on synchronization mediated by inhibition: i) the firing rate response to the noisy background input, ii) the membrane potential distribution, and iii) the shape of Inhibitory Post-Synaptic Potentials (IPSPs). For hyperpolarizing inhibition, the GIF IPSP profile (factor iii)) exhibits post-inhibitory rebound, which induces a coherent spike-mediated depolarization across cells that greatly facilitates synchronous oscillations. This effect dominates the network dynamics, hence GIF networks display stronger oscillations than IF networks. However, the restorative current in the GIF neuron lowers firing rates and narrows the membrane potential distribution (factors i) and ii), respectively), which tend to decrease synchrony. If inhibition is shunting instead of hyperpolarizing, post-inhibitory rebound is not elicited and factors i) and ii) dominate, yielding lower synchrony in GIF networks than in IF networks.

Neurons in the brain engage in collective oscillations at different frequencies. Gamma and high-gamma oscillations (30–100 Hz and higher) have been associated with cognitive functions, and are altered in psychiatric disorders such as schizophrenia and autism. Our understanding of how high-frequency oscillations are orchestrated in the brain is still limited, but it is necessary for the development of effective clinical approaches to the treatment of these disorders. Some neuron types exhibit dynamical properties that can favour synchronization. The theory of weakly coupled oscillators showed how the phase response of individual neurons can predict the patterns of phase relationships that are observed at the network level. However, neurons in vivo do not behave like regular oscillators, but fire irregularly in a regime dominated by fluctuations. Hence, which intrinsic dynamical properties matter for synchronization, and in which regime, is still an open question. Here, we show how single-cell damped subthreshold oscillations enhance synchrony in interneuronal networks by introducing a depolarizing component, mediated by post-inhibitory rebound, that is correlated among neurons due to common inhibitory input.

Effect of heterogeneity and noise on cross frequency phase-phase and phase-amplitude coupling

Cross-frequency coupling is hypothesized to play a functional role in neural computation. We apply phase resetting theory to two types of cross-frequency coupling that can occur when a slower oscillator periodically forces one or more oscillators: phase-phase coupling, in which the two oscillations are phase-locked, and phase-amplitude coupling, in which the amplitude of the driven oscillation is modulated. Our first result is that the shape of the phase resetting curve predicts the tightness of locking to a pulsatile forcing periodic input at any ratio of forced to intrinsic period; the tightness of the locking decreases as the ratio increases. Theoretical expressions were obtained for the probability density of the phases for a population of heterogeneous oscillators or a noisy single oscillator. Results were confirmed using two types of simulated networks and experiments on hippocampal CA1 neurons. Theoretical expressions were also obtained and confirmed for the probability density of N spike times within a single cycle of low frequency forcing. The second result is a suggested mechanism for phase-amplitude coupling in which progressive desynchronization leads to decreasing amplitude during a low frequency forcing cycle. Network simulations confirmed the theoretical viability of this mechanism, and that it generalizes to more diffuse input.

Effect of phase response curve skew on synchronization with and without conduction delays

A central problem in cortical processing including sensory binding and attentional gating is how neurons can synchronize their responses with zero or near-zero time lag. For a spontaneously firing neuron, an input from another neuron can delay or advance the next spike by different amounts depending upon the timing of the input relative to the previous spike. This information constitutes the phase response curve (PRC). We present a simple graphical method for determining the effect of PRC shape on synchronization tendencies and illustrate it using type 1 PRCs, which consist entirely of advances (delays) in response to excitation (inhibition). We obtained the following generic solutions for type 1 PRCs, which include the pulse-coupled leaky integrate and fire model. For pairs with mutual excitation, exact synchrony can be stable for strong coupling because of the stabilizing effect of the causal limit region of the PRC in which an input triggers a spike immediately upon arrival. However, synchrony is unstable for short delays, because delayed inputs arrive during a refractory period and cannot trigger an immediate spike. Right skew destabilizes antiphase and enables modes with time lags that grow as the conduction delay is increased. Therefore, right skew favors near synchrony at short conduction delays and a gradual transition between synchrony and antiphase for pairs coupled by mutual excitation. For pairs with mutual inhibition, zero time lag synchrony is stable for conduction delays ranging from zero to a substantial fraction of the period for pairs. However, for right skew there is a preferred antiphase mode at short delays. In contrast to mutual excitation, left skew destabilizes antiphase for mutual inhibition so that synchrony dominates at short delays as well. These pairwise synchronization tendencies constrain the synchronization properties of neurons embedded in larger networks.

Impact of neuronal heterogeneity on correlated colored noise-induced synchronization

Synchronization plays an important role in neural signal processing and transmission. Many hypotheses have been proposed to explain the origin of neural synchronization. In recent years, correlated noise-induced synchronization has received support from many theoretical and experimental studies. However, many of these prior studies have assumed that neurons have identical biophysical properties and that their inputs are well modeled by white noise. In this context, we use colored noise to induce synchronization between oscillators with heterogeneity in both phase-response curves and frequencies. In the low noise limit, we derive novel analytical theory showing that the time constant of colored noise influences correlated noise-induced synchronization and that oscillator heterogeneity can limit synchronization. Surprisingly, however, heterogeneous oscillators may synchronize better than homogeneous oscillators given low input correlations. We also find resonance of oscillator synchronization to colored noise inputs when firing frequencies diverge. Collectively, these results prove robust for both relatively high noise regimes and when applied to biophysically realistic spiking neuron models, and further match experimental recordings from acute brain slices.

Spike width and frequency alter stability of phase-locking in electrically coupled neurons

The stability of phase-locked states of electrically coupled type-1 phase response curve neurons is studied using piecewise linear formulations for their voltage profile and phase response curves. We find that at low frequency and/or small spike width, synchrony is stable, and antisynchrony unstable. At high frequency and/or large spike width, these phase-locked states switch their stability. Increasing the ratio of spike width to spike height causes the antisynchronous state to transition into a stable synchronous state. We compute the interaction function and the boundaries of stability of both these phase-locked states, and present analytical expressions for them. We also study the effect of phase response curve skewness on the boundaries of synchrony and antisynchrony.

Phase precession of grid cells in a network model without external pacemaker

Rodent brains encode space in both the firing rate and the spike timing of neurons in the medial entorhinal cortex. The rate code is realized by grid fields, that is, the neurons fire at multiple places that are arranged on a hexagonal lattice. Such activity is accompanied by theta oscillations of the local field potential. The phase of spikes thereby encodes space as well, since it decreases with the distance traveled in the field-a phenomenon called phase precession. A likely candidate for grid cells are entorhinal cortex stellate cells, which are type II oscillators and have been suggested to act as pacemakers. It is unclear how spiking of such putative pacemaker neurons would be able to precess in phase relative to a self-generated oscillation. This article presents a computational model of how this paradox can be resolved although the periodicity of the grid fields interferes with the periodic firing of the neurons. Our simulations show that the connections between stellate cells synchronize small cell groups, which allows a population oscillation during grid field activity that is accompanied by theta phase precession. Direct excitatory coupling between the stellate cells, indirect inhibitory coupling via a gamma-oscillating network of interneurons, or both could mediate this phase coordination. Our model further suggests modulation of h-currents to be a feasible mechanism to adjust phase precession to running-speed. The coexistence of rate and timing code for space hence follows as a natural consequence of the self-organization in a recurrent network.

A dynamical role for acetylcholine in synaptic renormalization

Although sleep is a fundamental behavior observed in virtually all animal species, its functions remain unclear. One leading proposal, known as the synaptic renormalization hypothesis, suggests that sleep is necessary to counteract a global strengthening of synapses that occurs during wakefulness. Evidence for sleep-dependent synaptic downscaling (or synaptic renormalization) has been observed experimentally, but the physiological mechanisms which generate this phenomenon are unknown. In this study, we propose that changes in neuronal membrane excitability induced by acetylcholine may provide a dynamical mechanism for both wake-dependent synaptic upscaling and sleep-dependent downscaling. We show in silico that cholinergically-induced changes in network firing patterns alter overall network synaptic potentiation when synaptic strengths evolve through spike-timing dependent plasticity mechanisms. Specifically, network synaptic potentiation increases dramatically with high cholinergic concentration and decreases dramatically with low levels of acetylcholine. We demonstrate that this phenomenon is robust across variation of many different network parameters.

Parameterized phase response curves for characterizing neuronal behaviors under transient conditions

Phase response curves (PRCs) are a simple model of how a neuron's spike time is affected by synaptic inputs. PRCs are useful in predicting how networks of neurons behave when connected. One challenge in estimating a neuron's PRCs experimentally is that many neurons do not have stationary firing rates. In this article we introduce a new method to estimate PRCs as a function of firing rate of the neuron. We call the resulting model a parameterized PRC (pPRC). Experimentally, we perturb the neuron applying a current with two parts: 1) a current held constant between spikes but changed at the onset of a spike, used to make the neuron fire at different rates, and 2) a pulse to emulate a synaptic input. A model of the applied constant current and the history is made to predict the interspike interval (ISI). A second model is then made to fit the modulation of the spike time from the expected ISI by the pulsatile stimulus. A polynomial with two independent variables, the stimulus phase and the expected ISI, is used to model the pPRC. The pPRC is validated in a computational model and applied to pyramidal neurons from the CA1 region of the hippocampal slices from rat. The pPRC can be used to model the effect of changing firing rates on network synchrony. It can also be used to characterize the effects of neuromodulators and genetic mutations (among other manipulations) on network synchrony. It can also easily be extended to account for more variables.

Moving beyond Type I and Type II neuron types

In 1948, Hodgkin delineated different classes of axonal firing.  This has been mathematically translated allowing insight and understanding to emerge.  As such, the terminology of ‘Type I’ and ‘Type II’ neurons is commonplace in the Neuroscience literature today.  Theoretical insights have helped us realize that, for example, network synchronization depends on whether neurons are Type I or Type II.  Mathematical models are precise with analyses (considering Type I/II aspects), but experimentally, the distinction can be less clear.  On the other hand, experiments are becoming more sophisticated in terms of distinguishing and manipulating particular cell types but are limited in terms of being able to consider network aspects simultaneously.   Although there is much work going on mathematically and experimentally, in my opinion it is becoming common that models are either superficially linked with experiment or not described in enough detail to appreciate the biological context.  Overall, we all suffer in terms of impeding our understanding of brain networks and applying our understanding to neurological disease.  I suggest that more modelers become familiar with experimental details and that more experimentalists appreciate modeling assumptions. In other words, we need to move beyond our comfort zones.

The robustness of phase-locking in neurons with dendro-dendritic electrical coupling

We examine the effects of dendritic filtering on the existence, stability, and robustness of phase-locked states to heterogeneity and noise in a pair of electrically coupled ball-and-stick neurons with passive dendrites. We use the theory of weakly coupled oscillators and analytically derived filtering properties of the dendritic coupling to systematically explore how the electrotonic length and diameter of dendrites can alter phase-locking. In the case of a fixed value of the coupling conductance (gc) taken from the literature, we find that repeated exchanges in stability between the synchronous and anti-phase states can occur as the electrical coupling becomes more distally located on the dendrites. However, the robustness of the phase-locked states in this case decreases rapidly towards zero as the distance between the electrical coupling and the somata increases. Published estimates of gc are calculated from the experimentally measured coupling coefficient (CC) based on a single-compartment description of a neuron, and therefore may be severe underestimates of gc. With this in mind, we re-examine the stability and robustness of phase-locking using a fixed value of CC, which imposes a limit on the maximum distance the electrical coupling can be located away from the somata. In this case, although the phase-locked states remain robust over the entire range of possible coupling locations, no exchanges in stability with changing coupling position are observed except for a single exchange that occurs in the case of a high somatic firing frequency and a large dendritic radius. Thus, our analysis suggests that multiple exchanges in stability with changing coupling location are unlikely to be observed in real neural systems.

Intrinsic heterogeneity in oscillatory dynamics limits correlation-induced neural synchronization

Synchronous neural oscillations are found throughout the brain and are thought to contribute to neural coding and the propagation of activity. Several proposed mechanisms of synchronization have gained support through combined theoretical and experimental investigation, including mechanisms based on coupling and correlated input. Here, we ask how correlation-induced synchrony is affected by physiological heterogeneity across neurons. To address this question, we examined cell-to-cell differences in phase-response curves (PRCs), which characterize the response of periodically firing neurons to weak perturbations. Using acute slice electrophysiology, we measured PRCs across a single class of principal neurons capable of sensory-evoked oscillations in vivo: the olfactory bulb mitral cells (MCs). Periodically firing MCs displayed a broad range of PRCs, each of which was well fit by a simple three-parameter model. MCs also displayed differences in firing rate-current relationships and in preferred firing rate ranges. Both the observed PRC heterogeneity and moderate firing rate differences (∼10 Hz) separately reduced the maximum correlation-induced synchrony between MCs by up to 25-30%. Simulations further demonstrated that these components of heterogeneity alone were sufficient to account for the difference in synchronization among heterogeneous vs. homogeneous populations in vitro. Within this simulation framework, independent modulation of specific PRC features additionally revealed which aspects of PRC heterogeneity most strongly impact correlation-induced synchronization. Finally, we demonstrated good agreement of novel mathematical theory with our experimental and simulation results, providing a theoretical basis for the influence of heterogeneity on correlation-induced neural synchronization.

Impact of adaptation currents on synchronization of coupled exponential integrate-and-fire neurons

The ability of spiking neurons to synchronize their activity in a network depends on the response behavior of these neurons as quantified by the phase response curve (PRC) and on coupling properties. The PRC characterizes the effects of transient inputs on spike timing and can be measured experimentally. Here we use the adaptive exponential integrate-and-fire (aEIF) neuron model to determine how subthreshold and spike-triggered slow adaptation currents shape the PRC. Based on that, we predict how synchrony and phase locked states of coupled neurons change in presence of synaptic delays and unequal coupling strengths. We find that increased subthreshold adaptation currents cause a transition of the PRC from only phase advances to phase advances and delays in response to excitatory perturbations. Increased spike-triggered adaptation currents on the other hand predominantly skew the PRC to the right. Both adaptation induced changes of the PRC are modulated by spike frequency, being more prominent at lower frequencies. Applying phase reduction theory, we show that subthreshold adaptation stabilizes synchrony for pairs of coupled excitatory neurons, while spike-triggered adaptation causes locking with a small phase difference, as long as synaptic heterogeneities are negligible. For inhibitory pairs synchrony is stable and robust against conduction delays, and adaptation can mediate bistability of in-phase and anti-phase locking. We further demonstrate that stable synchrony and bistable in/anti-phase locking of pairs carry over to synchronization and clustering of larger networks. The effects of adaptation in aEIF neurons on PRCs and network dynamics qualitatively reflect those of biophysical adaptation currents in detailed Hodgkin-Huxley-based neurons, which underscores the utility of the aEIF model for investigating the dynamical behavior of networks. Our results suggest neuronal spike frequency adaptation as a mechanism synchronizing low frequency oscillations in local excitatory networks, but indicate that inhibition rather than excitation generates coherent rhythms at higher frequencies.

Synchronization of neuronal spiking in the brain is related to cognitive functions, such as perception, attention, and memory. It is therefore important to determine which properties of neurons influence their collective behavior in a network and to understand how. A prominent feature of many cortical neurons is spike frequency adaptation, which is caused by slow transmembrane currents. We investigated how these adaptation currents affect the synchronization tendency of coupled model neurons. Using the efficient adaptive exponential integrate-and-fire (aEIF) model and a biophysically detailed neuron model for validation, we found that increased adaptation currents promote synchronization of coupled excitatory neurons at lower spike frequencies, as long as the conduction delays between the neurons are negligible. Inhibitory neurons on the other hand synchronize in presence of conduction delays, with or without adaptation currents. Our results emphasize the utility of the aEIF model for computational studies of neuronal network dynamics. We conclude that adaptation currents provide a mechanism to generate low frequency oscillations in local populations of excitatory neurons, while faster rhythms seem to be caused by inhibition rather than excitation.

Dynamical changes in neurons during seizures determine tonic to clonic shift

A tonic-clonic seizure transitions from high frequency asynchronous activity to low frequency coherent oscillations, yet the mechanism of transition remains unknown. We propose a shift in network synchrony due to changes in cellular response. Here we use phase-response curves (PRC) from Morris-Lecar (M-L) model neurons with synaptic depression and gradually decrease input current to cells within a network simulation. This method effectively decreases firing rates resulting in a shift to greater network synchrony illustrating a possible mechanism of the transition phenomenon. PRCs are measured from the M-L conductance based model cell with a range of input currents within the limit cycle. A large network of 3000 excitatory neurons is simulated with a network topology generated from second-order statistics which allows a range of population synchrony. The population synchrony of the oscillating cells is measured with the Kuramoto order parameter, which reveals a transition from tonic to clonic phase exhibited by our model network. The cellular response shift mechanism for the tonic-clonic seizure transition reproduces the population behavior closely when compared to EEG data.