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.     

State transitions through inhibitory interneurons in a cortical network model

Inhibitory interneurons shape the spiking characteristics and computational properties of cortical networks. Interneuron subtypes can precisely regulate cortical function but the roles of interneuron subtypes for promoting different regimes of cortical activity remains unclear. Therefore, we investigated the impact of fast spiking and non-fast spiking interneuron subtypes on cortical activity using a network model with connectivity and synaptic properties constrained by experimental data. We found that network properties were more sensitive to modulation of the fast spiking population, with reductions of fast spiking excitability generating strong spike correlations and network oscillations. Paradoxically, reduced fast spiking excitability produced a reduction of global excitation-inhibition balance and features of an inhibition stabilised network, in which firing rates were driven by the activity of excitatory neurons within the network. Further analysis revealed that the synaptic interactions and biophysical features associated with fast spiking interneurons, in particular their rapid intrinsic response properties and short synaptic latency, enabled this state transition by enhancing gain within the excitatory population. Therefore, fast spiking interneurons may be uniquely positioned to control the strength of recurrent excitatory connectivity and the transition to an inhibition stabilised regime. Overall, our results suggest that interneuron subtypes can exert selective control over excitatory gain allowing for differential modulation of global network state.     

Clustering in small networks of excitatory neurons with heterogeneous coupling strengths

Excitatory coupling with a slow rise time destabilizes synchrony between coupled neurons. Thus, the fully synchronous state is usually unstable in networks of excitatory neurons. Phase-clustered states, in which neurons are divided into multiple synchronized clusters, have also been found unstable in numerical studies of excitatory networks in the presence of noise. The question arises as to whether synchrony is possible in networks of neurons coupled through slow, excitatory synapses. In this paper, we show that robust, synchronous clustered states can occur in such networks. The effects of non-uniform distributions of coupling strengths are explored. Conditions for the existence and stability of clustered states are derived analytically. The analysis shows that a multi-cluster state can be stable in excitatory networks if the overall interactions between neurons in different clusters are stabilizing and strong enough to counter-act the destabilizing interactions between neurons within each cluster. When heterogeneity in the coupling strengths strengthens the stabilizing inter-cluster interactions and/or weakens the destabilizing in-cluster interactions, robust clustered states can occur in excitatory networks of all known model neurons. Numerical simulations were carried out to support the analytical results.     

Synchronization and Oscillatory Dynamics in Heterogeneous, Mutually Inhibited Neurons

We study some mechanisms responsible for synchronous oscillations and loss of synchrony at physiologically relevant frequencies (10–200 Hz) in a network of heterogeneous inhibitory neurons. We focus on the factors that determine the level of synchrony and frequency of the network response, as well as the effects of mild heterogeneity on network dynamics. With mild heterogeneity, synchrony is never perfect and is relatively fragile. In addition, the effects of inhibition are more complex in mildly heterogeneous networks than in homogeneous ones. In the former, synchrony is broken in two distinct ways, depending on the ratio of the synaptic decay time to the period of repetitive action potentials (_s/T), where T can be determined either from the network or from a single, self-inhibiting neuron. With _s/T > 2, corresponding to large applied current, small synaptic strength or large synaptic decay time, the effects of inhibition are largely tonic and heterogeneous neurons spike relatively independently. With _s/T < 1, synchrony breaks when faster cells begin to suppress their less excitable neighbors; cells that fire remain nearly synchronous. We show numerically that the behavior of mildly heterogeneous networks can be related to the behavior of single, self-inhibiting cells, which can be studied analytically.     

Alpha-Frequency Rhythms Desynchronize over Long Cortical Distances: A Modeling Study

Neocortical networks of excitatory and inhibitory neurons can display alpha()-frequency rhythms when an animal is in a resting or unfocused state. Unlike some - and -frequency rhythms, experimental observations in cats have shown that these -frequency rhythms need not synchronize over long cortical distances. Here, we develop a network model of synaptically coupled excitatory and inhibitory cells to study this asynchrony. The cells of the local circuit are modeled on the neurons found in layer V of the neocortex where -frequency rhythms are thought to originate. Cortical distance is represented by a pair of local circuits coupled with a delay in synaptic propagation. Mathematical analysis of this model reveals that the h and T currents present in layer V pyramidal (excitatory) cells not only produce and regulate the -frequency rhythm but also lead to the occurrence of spatial asynchrony. In particular, these inward currents cause excitation and inhibition to have nonintuitive effects in the network, with excitation delaying and inhibition advancing the firing time of cells; these reversed effects create the asynchrony. Moreover, increased excitatory to excitatory connections can lead to further desynchronization. However, the local rhythms have the property that, in the absence of excitatory to excitatory connections, if the participating cells are brought close to synchrony (for example, by common input), they will remain close to synchrony for a substantial time.     

Inhibitory postsynaptic potentials carry synchronized frequency information in active cortical networks

Temporal precision in spike timing is important in cortical function, interactions, and plasticity. We found that, during periods of recurrent network activity (UP states), cortical pyramidal cells in vivo and in vitro receive strong barrages of both excitatory and inhibitory postsynaptic potentials, with the inhibitory potentials showing much higher power at all frequencies above approximately 10 Hz and more synchrony between nearby neurons. Fast-spiking inhibitory interneurons discharged strongly in relation to higher-frequency oscillations in the field potential in vivo and possess membrane, synaptic, and action potential properties that are advantageous for transmission of higher-frequency activity. Intracellular injection of synaptic conductances having the characteristics of the recorded EPSPs and IPSPs reveal that IPSPs are important in controlling the timing and probability of action potential generation in pyramidal cells. Our results support the hypothesis that inhibitory networks are largely responsible for the dissemination of higher-frequency activity in cortex.     

Surprising Effects of Synaptic Excitation

Typically, excitatory synaptic coupling is thought of as an influence that accelerates and propagates firing in neuronal networks. This paper reviews recent results explaining how, contrary to these expectations, the presence of excitatory synaptic coupling can drastically slow oscillations in a network and how localized, sustained activity can arise in a network with purely excitatory coupling, without sustained inputs. These two effects stem from interactions of excitatory coupling with two different forms of intrinsic neuronal dynamics, and both serve to highlight the fact that the influence of synaptic coupling in a network depends strongly on the intrinsic properties of cells in the network.This work was partially supported by the National Science Foundation, under award DMS-0414023     

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

Spatiotemporal pattern formation in neuronal networks depends on the interplay between cellular and network synchronization properties. The neuronal phase response curve (PRC) is an experimentally obtainable measure that characterizes the cellular response to small perturbations, and can serve as an indicator of cellular propensity for synchronization. Two broad classes of PRCs have been identified for neurons: Type I, in which small excitatory perturbations induce only advances in firing, and Type II, in which small excitatory perturbations can induce both advances and delays in firing. Interestingly, neuronal PRCs are usually attenuated with increased spiking frequency, and Type II PRCs typically exhibit a greater attenuation of the phase delay region than of the phase advance region. We found that this phenomenon arises from an interplay between the time constants of active ionic currents and the interspike interval. As a result, excitatory networks consisting of neurons with Type I PRCs responded very differently to frequency modulation compared to excitatory networks composed of neurons with Type II PRCs. Specifically, increased frequency induced a sharp decrease in synchrony of networks of Type II neurons, while frequency increases only minimally affected synchrony in networks of Type I neurons. These results are demonstrated in networks in which both types of neurons were modeled generically with the Morris-Lecar model, as well as in networks consisting of Hodgkin-Huxley-based model cortical pyramidal cells in which simulated effects of acetylcholine changed PRC type. These results are robust to different network structures, synaptic strengths and modes of driving neuronal activity, and they indicate that Type I and Type II excitatory networks may display two distinct modes of processing information.     

Synchronous Bursting Can Arise from Mutual Excitation, Even When Individual Cells are not Endogenous Bursters

Mutual excitation between two neurons is generally thought to raise the excitation level of each neuron or, if they are both bursty, to act to synchronize their bursts. If only one is bursty, it can induce synchronized bursts in the other cell. Here we show that two nonbursty cells can be induced to burst in synchrony by mutual excitatory synaptic connections, provided the presynaptic threshold for graded synaptic transmission ateach synapse is at a different level. This mechanism may operate in a recently discovered network in the lobster Homarus gammarus.By a duality between presynaptic threshold and injected current, we also show that two identical, nonbursty, mutual excitatory cells could be induced to burst in synchrony by injecting differing amounts of current inthe two cells. Finally we show that differential oscillations betweentwo mutual excitatory cells could be stopped by a slow-tailedhyperpolarizing current pulse into one cell or a slow-taileddepolarizing pulse into the other.     

Inhibitory interneuron deficit links altered network activity and cognitive dysfunction in Alzheimer model

Alzheimer's disease (AD) results in cognitive decline and altered network activity, but the mechanisms are?unknown. We studied human amyloid precursor protein (hAPP) transgenic mice, which simulate key aspects of AD. Electroencephalographic recordings in hAPP mice revealed spontaneous epileptiform discharges, indicating network hypersynchrony, primarily during reduced gamma oscillatory activity. Because this oscillatory rhythm is generated by inhibitory parvalbumin (PV) cells, network dysfunction in hAPP mice might arise from impaired PV cells. Supporting this hypothesis, hAPP mice and AD patients had decreased levels of the interneuron-specific and PV cell-predominant voltage-gated sodium channel subunit Nav1.1. Restoring Nav1.1 levels in hAPP mice by Nav1.1-BAC expression increased inhibitory synaptic activity and gamma oscillations and reduced hypersynchrony, memory deficits, and premature mortality. We conclude that reduced Nav1.1 levels and PV cell dysfunction critically contribute to abnormalities in oscillatory rhythms, network synchrony, and memory in hAPP mice and possibly in AD.     

Emergence and fragmentation of the alpha-band driven by neuronal network dynamics

Rhythmic neuronal network activity underlies brain oscillations. To investigate how connected neuronal networks contribute to the emergence of the α-band and to the regulation of Up and Down states, we study a model based on synaptic short-term depression-facilitation with afterhyperpolarization (AHP). We found that the α-band is generated by the network behavior near the attractor of the Up-state. Coupling inhibitory and excitatory networks by reciprocal connections leads to the emergence of a stable α-band during the Up states, as reflected in the spectrogram. To better characterize the emergence and stability of thalamocortical oscillations containing α and δ rhythms during anesthesia, we model the interaction of two excitatory networks with one inhibitory network, showing that this minimal topology underlies the generation of a persistent α-band in the neuronal voltage characterized by dominant Up over Down states. Finally, we show that the emergence of the α-band appears when external inputs are suppressed, while fragmentation occurs at small synaptic noise or with increasing inhibitory inputs. To conclude, α-oscillations could result from the synaptic dynamics of interacting excitatory neuronal networks with and without AHP, a principle that could apply to other rhythms.     

The locust olfactory system as a case study for modeling dynamics of neurobiological networks: from discrete time neurons to continuous time neurons

Both chaotic and periodic activities are observed in networks of the central nervous systems. We choose the locust olfactory system as a good case study to analyze the relationships between networks' structure and the types of dynamics involved in coding mechanisms. In our modeling approach, we first build a fully connected recurrent network of synchronously updated McCulloch and Pitts neurons (MC-P type). In order to measure the use of the temporal dimension in the complex spatio-temporal patterns produced by the networks, we have defined an index the Normalized Euclidian Distance NED. We find that for appropriate parameters of input and connectivity, when adding some strong connections to the initial random synaptic matrices, it was easy to get the emergence of both robust oscillations and distributed synchrony in the spatiotemporal patterns. Then, in order to validate the MC-P model as a tool for analysis for network properties, we examine the dynamic behavior of networks of continuous time model neuron (Izhikevitch Integrate and Fire model -IFI-), implementing the same network characteristics. In both models, similarly to biological PN, the activity of excitatory neurons are phase-locked to different cycles of oscillations which remind the ones of the local field potential (LFP), and nevertheless exhibit dynamic behavior complex enough to be the basis of spatio-temporal codes.     

A neural network model of reliably optimized spike transmission

We studied the detailed structure of a neuronal network model in which the spontaneous spike activity is correctly optimized to match the experimental data and discuss the reliability of the optimized spike transmission. Two stochastic properties of the spontaneous activity were calculated: the spike-count rate and synchrony size. The synchrony size, expected to be an important factor for optimization of spike transmission in the network, represents a percentage of observed coactive neurons within a time bin, whose probability approximately follows a power-law. We systematically investigated how these stochastic properties could matched to those calculated from the experimental data in terms of the log-normally distributed synaptic weights between excitatory and inhibitory neurons and synaptic background activity induced by the input current noise in the network model. To ensure reliably optimized spike transmission, the synchrony size as well as spike-count rate were simultaneously optimized. This required changeably balanced log-normal distributions of synaptic weights between excitatory and inhibitory neurons and appropriately amplified synaptic background activity. Our results suggested that the inhibitory neurons with a hub-like structure driven by intensive feedback from excitatory neurons were a key factor in the simultaneous optimization of the spike-count rate and synchrony size, regardless of different spiking types between excitatory and inhibitory neurons.     

Membrane properties and the balance between excitation and inhibition control gamma-frequency oscillations arising from feedback inhibition

Computational studies as well as in vivo and in vitro results have shown that many cortical neurons fire in a highly irregular manner and at low average firing rates. These patterns seem to persist even when highly rhythmic signals are recorded by local field potential electrodes or other methods that quantify the summed behavior of a local population. Models of the 30-80 Hz gamma rhythm in which network oscillations arise through 'stochastic synchrony' capture the variability observed in the spike output of single cells while preserving network-level organization. We extend upon these results by constructing model networks constrained by experimental measurements and using them to probe the effect of biophysical parameters on network-level activity. We find in simulations that gamma-frequency oscillations are enabled by a high level of incoherent synaptic conductance input, similar to the barrage of noisy synaptic input that cortical neurons have been shown to receive in vivo. This incoherent synaptic input increases the emergent network frequency by shortening the time scale of the membrane in excitatory neurons and by reducing the temporal separation between excitation and inhibition due to decreased spike latency in inhibitory neurons. These mechanisms are demonstrated in simulations and in vitro current-clamp and dynamic-clamp experiments. Simulation results further indicate that the membrane potential noise amplitude has a large impact on network frequency and that the balance between excitatory and inhibitory currents controls network stability and sensitivity to external inputs.     

Emergent oscillations in a realistic network: the role of inhibition and the effect of the spatiotemporal distribution of the input.

We have simulated a network of 10,000 two-compartment cells, spatially distributed on a two-dimensional sheet; 15% of the cells were inhibitory. The input to the network was spatially delimited. Global oscillations frequently were achieved with a simple set of connectivity rules. The inhibitory neurons paced the network, whereas the excitatory neurons amplified the input, permitting oscillations at low-input intensities. Inhibitory neurons were active over a greater area than excitatory ones, forming a ring of inhibition. The oscillation frequency was modulated to some extent by the input intensity, as has been shown experimentally in the striate cortex, but predominantly by the properties of the inhibitory neurons and their connections: the membrane and synaptic time constants and the distribution of delays. In networks that showed oscillations and in those that did not, widely distributed inputs could lead to the specific recruitment of the inhibitory neurons and to near zero activity of the excitatory cells. Hence the spatial distribution of excitatory inputs could provide a means of selectively exciting or inhibiting a target network. Finally, neither the presence of oscillations nor the global spike activity provided any reliable indication of the level of excitatory output from the network.     

Emergent Oscillations in a Realistic Network: The Role of Inhibition and the Effect of the Spatiotemporal Distribution of the Input

We have simulated a network of 10,000 two-compartment cells, spatially distributed on a two-dimensional sheet; 15% of the cells were inhibitory. The input to the network was spatially delimited. Global oscillations frequently were achieved with a simple set of connectivity rules. The inhibitory neurons paced the network, whereas the excitatory neurons amplified the input, permitting oscillations at low-input intensities. Inhibitory neurons were active over a greater area than excitatory ones, forming a ring of inhibition. The oscillation frequency was modulated to some extent by the input intensity, as has been shown experimentally in the striate cortex, but predominantly by the properties of the inhibitory neurons and their connections: the membrane and synaptic time constants and the distribution of delays.In networks that showed oscillations and in those that did not, widely distributed inputs could lead to the specific recruitment of the inhibitory neurons and to near zero activity of the excitatory cells. Hence the spatial distribution of excitatory inputs could provide a means of selectively exciting or inhibiting a target network. Finally, neither the presence of oscillations nor the global spike activity provided any reliable indication of the level of excitatory output from the network.     

Phase resetting reduces theta–gamma rhythmic interaction to a one-dimensional map

Gamma and theta oscillations of the hippocampus are known to interact, but the mechanisms underlying such interaction are not well understood. We focus on a previously published computational model of hippocampal activity that shows the gamma rhythms nesting in the theta rhythms, and investigate the dynamical mechanisms underlying that interaction. There are three types of neurons in the model: pyramidal cells, fast-spiking interneurons, and “oriens lacunosum-moelculare” (O-LM cells); the latter is an inhibitory cell whose inhibition has a longer time scale, and which has currents associated with intrinsic theta-rhythm behavior. We identify two main modes of interaction among the slow and the fast rhythms in the model, modulated by the strength of the excitatory synapse on the O-LM cells. Using resets of phases after each pyramidal cell and O-LM spike, we extend the use of the phase transition map (PTM) to encode the stability type of spiking patterns in networks where different frequencies interact. The tailored application of the PTM to the model network measures how the interaction between the shape of the phase response curves and the length of the gamma period determines the number of gamma spikes in theta cycles, and provides an explicit formula for the length of theta intervals in nesting regimes. Using the PTM, we also explain the covariance of the gamma and theta rhythms as drive is changed over some intervals.     

Modeling the emergence of circadian rhythms in a clock neuron network

Circadian rhythms in pacemaker cells persist for weeks in constant darkness, while in other types of cells the molecular oscillations that underlie circadian rhythms damp rapidly under the same conditions. Although much progress has been made in understanding the biochemical and cellular basis of circadian rhythms, the mechanisms leading to damped or self-sustained oscillations remain largely unknown. There exist many mathematical models that reproduce the circadian rhythms in the case of a single cell of the Drosophila fly. However, not much is known about the mechanisms leading to coherent circadian oscillation in clock neuron networks. In this work we have implemented a model for a network of interacting clock neurons to describe the emergence (or damping) of circadian rhythms in Drosophila fly, in the absence of zeitgebers. Our model consists of an array of pacemakers that interact through the modulation of some parameters by a network feedback. The individual pacemakers are described by a well-known biochemical model for circadian oscillation, to which we have added degradation of PER protein by light and multiplicative noise. The network feedback is the PER protein level averaged over the whole network. In particular, we have investigated the effect of modulation of the parameters associated with (i) the control of net entrance of PER into the nucleus and (ii) the non-photic degradation of PER. Our results indicate that the modulation of PER entrance into the nucleus allows the synchronization of clock neurons, leading to coherent circadian oscillations under constant dark condition. On the other hand, the modulation of non-photic degradation cannot reset the phases of individual clocks subjected to intrinsic biochemical noise.     

Inhibitory synchrony as a mechanism for attentional gain modulation

Recordings from area V4 of monkeys have revealed that when the focus of attention is on a visual stimulus within the receptive field of a cortical neuron, two distinct changes can occur: The firing rate of the neuron can change and there can be an increase in the coherence between spikes and the local field potential (LFP) in the gamma-frequency range (30-50 Hz). The hypothesis explored here is that these observed effects of attention could be a consequence of changes in the synchrony of local interneuron networks. We performed computer simulations of a Hodgkin-Huxley type neuron driven by a constant depolarizing current, I, representing visual stimulation and a modulatory inhibitory input representing the effects of attention via local interneuron networks. We observed that the neuron's firing rate and the coherence of its output spike train with the synaptic inputs was modulated by the degree of synchrony of the inhibitory inputs. When inhibitory synchrony increased, the coherence of spiking model neurons with the synaptic input increased, but the firing rate either increased or remained the same. The mean number of synchronous inhibitory inputs was a key determinant of the shape of the firing rate versus current (f-I) curves. For a large number of inhibitory inputs (approximately 50), the f-I curve saturated for large I and an increase in input synchrony resulted in a shift of sensitivity-the model neuron responded to weaker inputs I. For a small number (approximately 10), the f-I curves were non-saturating and an increase in input synchrony led to an increase in the gain of the response-the firing rate in response to the same input was multiplied by an approximately constant factor. The firing rate modulation with inhibitory synchrony was highest when the input network oscillated in the gamma frequency range. Thus, the observed changes in firing rate and coherence of neurons in the visual cortex could be controlled by top-down inputs that regulated the coherence in the activity of a local inhibitory network discharging at gamma frequencies.     

Cellular, synaptic, network, and modulatory mechanisms involved in rhythm generation

The membrane properties and the synaptic interactions of individual neurons, as well as the interactions between neuronal networks, all contribute to the formation of the complex patterns of activity that underlie rhythmic motor patterns and slow-wave sleep rhythms. These properties and interactions are potential points of modulation for further refining network output. Recent work illustrates the range of these properties and interactions and suggests how they may be modulated.