Low-dimensional maps encoding dynamics in entorhinal cortex and hippocampus

Inter and intralaminar excitation of parvalbumin interneurons in mouse barrel cortex

Parvalbumin (PV) interneurons are inhibitory fast-spiking cells with essential roles in directing the flow of information through cortical circuits. These neurons set the balance between excitation and inhibition and control rhythmic activity. PV interneurons differ between cortical layers in their morphology, circuitry, and function, but how their electrophysiological properties vary has received little attention. Here we investigate responses of PV interneurons in different layers of primary somatosensory barrel cortex (BC) to different excitatory inputs. With the genetically-encoded hybrid voltage sensor, hVOS, we recorded voltage changes in many L2/3 and L4 PV interneurons simultaneously, with stimulation applied to either L2/3 or L4. A semi-automated procedure was developed to identify small regions of interest corresponding to single responsive PV interneurons. Amplitude, half-width, and rise-time were greater for PV interneurons residing in L2/3 compared to L4. Stimulation in L2/3 elicited responses in both L2/3 and L4 with longer latency compared to stimulation in L4. These differences in latency between layers could influence their windows for temporal integration. Thus, PV interneurons in different cortical layers of BC respond in a layer specific and input specific manner, and these differences have potential roles in cortical computations.

Inter and Intralaminar Excitation of Parvalbumin Interneurons in Mouse Barrel Cortex

Parvalbumin (PV) interneurons are inhibitory fast-spiking cells with essential roles in directing the flow of information through cortical circuits. These neurons set the balance between excitation and inhibition, control rhythmic activity, and have been linked to disorders including autism spectrum and schizophrenia. PV interneurons differ between cortical layers in their morphology, circuitry, and function, but how their electrophysiological properties vary has received little attention. Here we investigate responses of PV interneurons in different layers of primary somatosensory barrel cortex (BC) to different excitatory inputs. With the genetically-encoded hybrid voltage sensor, hVOS, we recorded voltage changes simultaneously in many L2/3 and L4 PV interneurons to stimulation in either L2/3 or L4. Decay-times were consistent across L2/3 and L4. Amplitude, half-width, and rise-time were greater for PV interneurons residing in L2/3 compared to L4. Stimulation in L2/3 elicited responses in both L2/3 and L4 with longer latency compared to stimulation in L4. These differences in latency between layers could influence their windows for temporal integration. Thus PV interneurons in different cortical layers of BC show differences in response properties with potential roles in cortical computations.

Interneuronal network model of theta-nested fast oscillations predicts differential effects of heterogeneity, gap junctions and short term depression for hyperpolarizing versus shunting inhibition

Theta and gamma oscillations in the hippocampus have been hypothesized to play a role in the encoding and retrieval of memories. Recently, it was shown that an intrinsic fast gamma mechanism in medial entorhinal cortex can be recruited by optogenetic stimulation at theta frequencies, which can persist with fast excitatory synaptic transmission blocked, suggesting a contribution of interneuronal network gamma (ING). We calibrated the passive and active properties of a 100-neuron model network to capture the range of passive properties and frequency/current relationships of experimentally recorded PV+ neurons in the medial entorhinal cortex (mEC). The strength and probabilities of chemical and electrical synapses were also calibrated using paired recordings, as were the kinetics and short-term depression (STD) of the chemical synapses. Gap junctions that contribute a noticeable fraction of the input resistance were required for synchrony with hyperpolarizing inhibition; these networks exhibited theta-nested high frequency oscillations similar to the putative ING observed experimentally in the optogenetically-driven PV-ChR2 mice. With STD included in the model, the network desynchronized at frequencies above ~200 Hz, so for sufficiently strong drive, fast oscillations were only observed before the peak of the theta. Because hyperpolarizing synapses provide a synchronizing drive that contributes to robustness in the presence of heterogeneity, synchronization decreases as the hyperpolarizing inhibition becomes weaker. In contrast, networks with shunting inhibition required non-physiological levels of gap junctions to synchronize using conduction delays within the measured range.

A Hypothesis for Theta Rhythm Frequency Control in CA1 Microcircuits

Computational models of neural circuits with varying levels of biophysical detail have been generated in pursuit of an underlying mechanism explaining the ubiquitous hippocampal theta rhythm. However, within the theta rhythm are at least two types with distinct frequencies associated with different behavioral states, an aspect that must be considered in pursuit of these mechanistic explanations. Here, using our previously developed excitatory-inhibitory network models that generate theta rhythms, we investigate the robustness of theta generation to intrinsic neuronal variability by building a database of heterogeneous excitatory cells and implementing them in our microcircuit model. We specifically investigate the impact of three key "building block" features of the excitatory cell model that underlie our model design: these cells' rheobase, their capacity for post-inhibitory rebound, and their spike-frequency adaptation. We show that theta rhythms at various frequencies can arise dependent upon the combination of these building block features, and we find that the speed of these oscillations are dependent upon the excitatory cells' response to inhibitory drive, as encapsulated by their phase response curves. Taken together, these findings support a hypothesis for theta frequency control that includes two aspects: (i) an internal mechanism that stems from the building block features of excitatory cell dynamics; (ii) an external mechanism that we describe as "inhibition-based tuning" of excitatory cell firing. We propose that these mechanisms control theta rhythm frequencies and underlie their robustness.

Large time step discrete-time modeling of sharp wave activity in hippocampal area CA3

Reduced models of neuronal spiking activity simulated with a fixed integration time are frequently used in studies of spatio-temporal dynamics of neurobiological networks. The choice of fixed time step integration provides computational simplicity and efficiency, especially in cases dealing with large number of neurons and synapses operating at a different level of activity across the population at any given time. A network model tuned to generate a particular type of oscillations or wave patterns is sensitive to the intrinsic properties of neurons and synapses and, therefore, commonly susceptible to changes the time step of integration. In this study, we analyzed a model of sharp-wave activity in the network of hippocampal area CA3, to examine how an increase of the integration time step affects network behavior and to propose adjustments of intrinsic properties neurons and synapses that help minimize or remove the damage caused by the time step increase.

Phase response theory explains cluster formation in sparsely but strongly connected inhibitory neural networks and effects of jitter due to sparse connectivity

We show how to predict whether a neural network will exhibit global synchrony (a one-cluster state) or a two-cluster state based on the assumption of pulsatile coupling and critically dependent upon the phase response curve (PRC) generated by the appropriate perturbation from a partner cluster. Our results hold for a monotonically increasing (meaning longer delays as the phase increases) PRC, which likely characterizes inhibitory fast-spiking basket and cortical low-threshold-spiking interneurons in response to strong inhibition. Conduction delays stabilize synchrony for this PRC shape, whereas they destroy two-cluster states, the former by avoiding a destabilizing discontinuity and the latter by approaching it. With conduction delays, stronger coupling strength can promote a one-cluster state, so the weak coupling limit is not applicable here. We show how jitter can destabilize global synchrony but not a two-cluster state. Local stability of global synchrony in an all-to-all network does not guarantee that global synchrony can be observed in an appropriately scaled sparsely connected network; the basin of attraction can be inferred from the PRC and must be sufficiently large. Two-cluster synchrony is not obviously different from one-cluster synchrony in the presence of noise and may be the actual substrate for oscillations observed in the local field potential (LFP) and the electroencephalogram (EEG) in situations where global synchrony is not possible. Transitions between cluster states may change the frequency of the rhythms observed in the LFP or EEG. Transitions between cluster states within an inhibitory subnetwork may allow more effective recruitment of pyramidal neurons into the network rhythm. NEW & NOTEWORTHY We show that jitter induced by sparse connectivity can destabilize global synchrony but not a two-cluster state with two smaller clusters firing alternately. On the other hand, conduction delays stabilize synchrony and destroy two-cluster states. These results hold if each cluster exhibits a phase response curve similar to one that characterizes fast-spiking basket and cortical low-threshold-spiking cells for strong inhibition. Either a two-cluster or a one-cluster state might provide the oscillatory substrate for neural computations.

Phase-locking and bistability in neuronal networks with synaptic depression

We consider a recurrent network of two oscillatory neurons that are coupled with inhibitory synapses. We use the phase response curves of the neurons and the properties of short-term synaptic depression to define Poincaré maps for the activity of the network. The fixed points of these maps correspond to phase-locked modes of the network. Using these maps, we analyze the conditions that allow short-term synaptic depression to lead to the existence of bistable phase-locked, periodic solutions. We show that bistability arises when either the phase response curve of the neuron or the short-term depression profile changes steeply enough. The results apply to any Type I oscillator and we illustrate our findings using the Quadratic Integrate-and-Fire and Morris-Lecar neuron models.

Predicting the Existence and Stability of Phase-Locked Mode in Neural Networks Using Generalized Phase-Resetting Curve

We used the phase-resetting method to study a biologically relevant three-neuron network in which one neuron receives multiple inputs per cycle. For this purpose, we first generalized the concept of phase resetting to accommodate multiple inputs per cycle. We explicitly showed how analytical conditions for the existence and the stability of phase-locked modes are derived. In particular, we solved newly derived recursive maps using as an example a biologically relevant driving-driven neural network with a dynamic feedback loop. We applied the generalized phase-resetting definition to predict the relative-phase and the stability of a phase-locked mode in open loop setup. We also compared the predicted phase-locked mode against numerical simulations of the fully connected network.

A generalized phase resetting method for phase-locked modes prediction

We derived analytically and checked numerically a set of novel conditions for the existence and the stability of phase-locked modes in a biologically relevant master-slave neural network with a dynamic feedback loop. Since neural oscillators even in the three-neuron network investigated here receive multiple inputs per cycle, we generalized the concept of phase resetting to accommodate multiple inputs per cycle. We proved that the phase resetting produced by two or more stimuli per cycle can be recursively computed from the traditional, single stimulus, phase resetting. We applied the newly derived generalized phase resetting definition to predicting the relative phase and the stability of a phase-locked mode that was experimentally observed in this type of master-slave network with a dynamic loop network.

Neural Cross-Frequency Coupling: Connecting Architectures, Mechanisms, and Functions

Neural oscillations are ubiquitously observed in the mammalian brain, but it has proven difficult to tie oscillatory patterns to specific cognitive operations. Notably, the coupling between neural oscillations at different timescales has recently received much attention, both from experimentalists and theoreticians. We review the mechanisms underlying various forms of this cross-frequency coupling. We show that different types of neural oscillators and cross-frequency interactions yield distinct signatures in neural dynamics. Finally, we associate these mechanisms with several putative functions of cross-frequency coupling, including neural representations of multiple environmental items, communication over distant areas, internal clocking of neural processes, and modulation of neural processing based on temporal predictions.

Neurosystems: brain rhythms and cognitive processing

Neuronal rhythms are ubiquitous features of brain dynamics, and are highly correlated with cognitive processing. However, the relationship between the physiological mechanisms producing these rhythms and the functions associated with the rhythms remains mysterious. This article investigates the contributions of rhythms to basic cognitive computations (such as filtering signals by coherence and/or frequency) and to major cognitive functions (such as attention and multi-modal coordination). We offer support to the premise that the physiology underlying brain rhythms plays an essential role in how these rhythms facilitate some cognitive operations.

Effects of synaptic plasticity on phase and period locking in a network of two oscillatory neurons

We study the effects of synaptic plasticity on the determination of firing period and relative phases in a network of two oscillatory neurons coupled with reciprocal inhibition. We combine the phase response curves of the neurons with the short-term synaptic plasticity properties of the synapses to define Poincaré maps for the activity of an oscillatory network. Fixed points of these maps correspond to the phase-locked modes of the network. These maps allow us to analyze the dependence of the resulting network activity on the properties of network components. Using a combination of analysis and simulations, we show how various parameters of the model affect the existence and stability of phase-locked solutions. We find conditions on the synaptic plasticity profiles and the phase response curves of the neurons for the network to be able to maintain a constant firing period, while varying the phase of locking between the neurons or vice versa. A generalization to cobwebbing for two-dimensional maps is also discussed.

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.

Short conduction delays cause inhibition rather than excitation to favor synchrony in hybrid neuronal networks of the entorhinal cortex

How stable synchrony in neuronal networks is sustained in the presence of conduction delays is an open question. The Dynamic Clamp was used to measure phase resetting curves (PRCs) for entorhinal cortical cells, and then to construct networks of two such neurons. PRCs were in general Type I (all advances or all delays) or weakly type II with a small region at early phases with the opposite type of resetting. We used previously developed theoretical methods based on PRCs under the assumption of pulsatile coupling to predict the delays that synchronize these hybrid circuits. For excitatory coupling, synchrony was predicted and observed only with no delay and for delays greater than half a network period that cause each neuron to receive an input late in its firing cycle and almost immediately fire an action potential. Synchronization for these long delays was surprisingly tight and robust to the noise and heterogeneity inherent in a biological system. In contrast to excitatory coupling, inhibitory coupling led to antiphase for no delay, very short delays and delays close to a network period, but to near-synchrony for a wide range of relatively short delays. PRC-based methods show that conduction delays can stabilize synchrony in several ways, including neutralizing a discontinuity introduced by strong inhibition, favoring synchrony in the case of noisy bistability, and avoiding an initial destabilizing region of a weakly type II PRC. PRCs can identify optimal conduction delays favoring synchronization at a given frequency, and also predict robustness to noise and heterogeneity.

Individual oscillators, such as pendulum-based clocks and fireflies, can spontaneously organize into a coherent, synchronized entity with a common frequency. Neurons can oscillate under some circumstances, and can synchronize their firing both within and across brain regions. Synchronized assemblies of neurons are thought to underlie cognitive functions such as recognition, recall, perception and attention. Pathological synchrony can lead to epilepsy, tremor and other dynamical diseases, and synchronization is altered in most mental disorders. Biological neurons synchronize despite conduction delays, heterogeneous circuit composition, and noise. In biological experiments, we built simple networks in which two living neurons could interact via a computer in real time. The computer precisely controlled the nature of the connectivity and the length of the communication delays. We characterized the synchronization tendencies of individual, isolated oscillators by measuring how much a single input delivered by the computer transiently shortened or lengthened the cycle period of the oscillation. We then used this information to correctly predict the strong dependence of the coordination pattern of the firing of the component neurons on the length of the communication delays. Upon this foundation, we can begin to build a theory of the basic principles of synchronization in more complex brain circuits.

Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators

Gamma oscillations can synchronize with near zero phase lag over multiple cortical regions and between hemispheres, and between two distal sites in hippocampal slices. How synchronization can take place over long distances in a stable manner is considered an open question. The phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike, depending upon where in the cycle it is received. We use PRCs under the assumption of pulsatile coupling to derive existence and stability criteria for 1:1 phase-locking that arises via bidirectional pulse coupling of two limit cycle oscillators with a conduction delay of any duration for any 1:1 firing pattern. The coupling can be strong as long as the effect of one input dissipates before the next input is received. We show the form that the generic synchronous and anti-phase solutions take in a system of two identical, identically pulse-coupled oscillators with identical delays. The stability criterion has a simple form that depends only on the slopes of the PRCs at the phases at which inputs are received and on the number of cycles required to complete the delayed feedback loop. The number of cycles required to complete the delayed feedback loop depends upon both the value of the delay and the firing pattern. We successfully tested the predictions of our methods on networks of model neurons. The criteria can easily be extended to include the effect of an input on the cycle after the one in which it is received.

Striatal origin of the pathologic beta oscillations in Parkinson's disease

Enhanced oscillations at beta frequencies (8-30 Hz) are a signature neural dynamic pathology in the basal ganglia and cortex of Parkinson's disease patients. The mechanisms underlying these pathological beta oscillations remain elusive. Here, using mathematical models, we find that robust beta oscillations can emerge from inhibitory interactions between striatal medium spiny neurons. The interaction of the synaptic GABAa currents and the intrinsic membrane M-current promotes population oscillations in the beta frequency range. Increased levels of cholinergic drive, a condition relevant to the parkinsonian striatum, lead to enhanced beta oscillations in the striatal model. We show experimentally that direct infusion of the cholinergic agonist carbachol into the striatum, but not into the neighboring cortex, of the awake, normal rodent induces prominent beta frequency oscillations in the local field potential. These results provide evidence for amplification of normal striatal network dynamics as a mechanism responsible for the enhanced beta frequency oscillations in Parkinson's disease.

Using computer simulations to determine the limitations of dynamic clamp stimuli applied at the soma in mimicking distributed conductance sources

In previous studies we used the technique of dynamic clamp to study how temporal modulation of inhibitory and excitatory inputs control the frequency and precise timing of spikes in neurons of the deep cerebellar nuclei (DCN). Although this technique is now widely used, it is limited to interpreting conductance inputs as being location independent; i.e., all inputs that are biologically distributed across the dendritic tree are applied to the soma. We used computer simulations of a morphologically realistic model of DCN neurons to compare the effects of purely somatic vs. distributed dendritic inputs in this cell type. We applied the same conductance stimuli used in our published experiments to the model. To simulate variability in neuronal responses to repeated stimuli, we added a somatic white current noise to reproduce subthreshold fluctuations in the membrane potential. We were able to replicate our dynamic clamp results with respect to spike rates and spike precision for different patterns of background synaptic activity. We found only minor differences in the spike pattern generation between focal or distributed input in this cell type even when strong inhibitory or excitatory bursts were applied. However, the location dependence of dynamic clamp stimuli is likely to be different for each cell type examined, and the simulation approach developed in the present study will allow a careful assessment of location dependence in all cell types.

The variance of phase-resetting curves

Phase resetting curves (PRCs) provide a measure of the sensitivity of oscillators to perturbations. In a noisy environment, these curves are themselves very noisy. Using perturbation theory, we compute the mean and the variance for PRCs for arbitrary limit cycle oscillators when the noise is small. Phase resetting curves and phase dependent variance are fit to experimental data and the variance is computed using an ad-hoc method. The theoretical curves of this phase dependent method match both simulations and experimental data significantly better than an ad-hoc method. A dual cell network simulation is compared to predictions using the analytical phase dependent variance estimation presented in this paper. We also discuss how entrainment of a neuron to a periodic pulse depends on the noise amplitude.

Phase-response curves and synchronized neural networks

We review the principal assumptions underlying the application of phase-response curves (PRCs) to synchronization in neuronal networks. The PRC measures how much a given synaptic input perturbs spike timing in a neural oscillator. Among other applications, PRCs make explicit predictions about whether a given network of interconnected neurons will synchronize, as is often observed in cortical structures. Regarding the assumptions of the PRC theory, we conclude: (i) The assumption of noise-tolerant cellular oscillations at or near the network frequency holds in some but not all cases. (ii) Reduced models for PRC-based analysis can be formally related to more realistic models. (iii) Spike-rate adaptation limits PRC-based analysis but does not invalidate it. (iv) The dependence of PRCs on synaptic location emphasizes the importance of improving methods of synaptic stimulation. (v) New methods can distinguish between oscillations that derive from mutual connections and those arising from common drive. (vi) It is helpful to assume linear summation of effects of synaptic inputs; experiments with trains of inputs call this assumption into question. (vii) Relatively subtle changes in network structure can invalidate PRC-based predictions. (viii) Heterogeneity in the preferred frequencies of component neurons does not invalidate PRC analysis, but can annihilate synchronous activity.

Stability of two cluster solutions in pulse coupled networks of neural oscillators

Phase response curves (PRCs) have been widely used to study synchronization in neural circuits comprised of pacemaking neurons. They describe how the timing of the next spike in a given spontaneously firing neuron is affected by the phase at which an input from another neuron is received. Here we study two reciprocally coupled clusters of pulse coupled oscillatory neurons. The neurons within each cluster are presumed to be identical and identically pulse coupled, but not necessarily identical to those in the other cluster. We investigate a two cluster solution in which all oscillators are synchronized within each cluster, but in which the two clusters are phase locked at nonzero phase with each other. Intuitively, one might expect this solution to be stable only when synchrony within each isolated cluster is stable, but this is not the case. We prove rigorously the stability of the two cluster solution and show how reciprocal coupling can stabilize synchrony within clusters that cannot synchronize in isolation. These stability results for the two cluster solution suggest a mechanism by which reciprocal coupling between brain regions can induce local synchronization via the network feedback loop.

Neurophysiological and computational principles of cortical rhythms in cognition

Synchronous rhythms represent a core mechanism for sculpting temporal coordination of neural activity in the brain-wide network. This review focuses on oscillations in the cerebral cortex that occur during cognition, in alert behaving conditions. Over the last two decades, experimental and modeling work has made great strides in elucidating the detailed cellular and circuit basis of these rhythms, particularly gamma and theta rhythms. The underlying physiological mechanisms are diverse (ranging from resonance and pacemaker properties of single cells to multiple scenarios for population synchronization and wave propagation), but also exhibit unifying principles. A major conceptual advance was the realization that synaptic inhibition plays a fundamental role in rhythmogenesis, either in an interneuronal network or in a reciprocal excitatory-inhibitory loop. Computational functions of synchronous oscillations in cognition are still a matter of debate among systems neuroscientists, in part because the notion of regular oscillation seems to contradict the common observation that spiking discharges of individual neurons in the cortex are highly stochastic and far from being clocklike. However, recent findings have led to a framework that goes beyond the conventional theory of coupled oscillators and reconciles the apparent dichotomy between irregular single neuron activity and field potential oscillations. From this perspective, a plethora of studies will be reviewed on the involvement of long-distance neuronal coherence in cognitive functions such as multisensory integration, working memory, and selective attention. Finally, implications of abnormal neural synchronization are discussed as they relate to mental disorders like schizophrenia and autism.

Pulse coupled oscillators and the phase resetting curve

Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators. In these oscillators, the interactions take the form of brief pulses such that the effect of one input dies out before the next is received. A phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. PRCs can be used to predict phase locking in networks of pulse coupled oscillators. In some studies of pulse coupled oscillators, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is generated using a perturbation that approximates the input received in the real biological network. In general, this approach requires that circuit architecture and a specific firing pattern be assumed. This allows the construction of discrete maps from one event to the next. The fixed points of these maps correspond to periodic firing modes and are easier to locate and analyze for stability compared to locating and analyzing periodic modes in the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks.

Cellular dynamical mechanisms for encoding the time and place of events along spatiotemporal trajectories in episodic memory

Understanding the mechanisms of episodic memory requires linking behavioral data and lesion effects to data on the dynamics of cellular membrane potentials and population interactions within brain regions. Linking behavior to specific membrane channels and neurochemicals has implications for therapeutic applications. Lesions of the hippocampus, entorhinal cortex and subcortical nuclei impair episodic memory function in humans and animals, and unit recording data from these regions in behaving animals indicate episodic memory processes. Intracellular recording in these regions demonstrates specific cellular properties including resonance, membrane potential oscillations and bistable persistent spiking that could underlie the encoding and retrieval of episodic trajectories. A model presented here shows how intrinsic dynamical properties of neurons could mediate the encoding of episodic memories as complex spatiotemporal trajectories. The dynamics of neurons allow encoding and retrieval of unique episodic trajectories in multiple continuous dimensions including temporal intervals, personal location, the spatial coordinates and sensory features of perceived objects and generated actions, and associations between these elements. The model also addresses how cellular dynamics could underlie unit firing data suggesting mechanisms for coding continuous dimensions of space, time, sensation and action.

Phase resetting curves allow for simple and accurate prediction of robust N:1 phase locking for strongly coupled neural oscillators

Existence and stability criteria for harmonic locking modes were derived for two reciprocally pulse coupled oscillators based on their first and second order phase resetting curves. Our theoretical methods are general in the sense that no assumptions about the strength of coupling, type of synaptic coupling, and model are made. These methods were then tested using two reciprocally inhibitory Wang and Buzsáki model neurons. The existence of bands of 2:1, 3:1, 4:1, and 5:1 phase locking in the relative frequency parameter space was predicted correctly, as was the phase of the slow neuron's spike within the cycle of the fast neuron in which it occurred. For weak coupling the bands are very narrow, but strong coupling broadens the bands. The predictions of the pulse coupled method agreed with weak coupling methods in the weak coupling regime, but extended predictability into the strong coupling regime. We show that our prediction method generalizes to pairs of neural oscillators coupled through excitatory synapses, and to networks of multiple oscillatory neurons. The main limitation of the method is the central assumption that the effect of each input dies out before the next input is received.

A phase code for memory could arise from circuit mechanisms in entorhinal cortex

Neurophysiological data reveals intrinsic cellular properties that suggest how entorhinal cortical neurons could code memory by the phase of their firing. Potential cellular mechanisms for this phase coding in models of entorhinal function are reviewed. This mechanism for phase coding provides a substrate for modeling the responses of entorhinal grid cells, as well as the replay of neural spiking activity during waking and sleep. Efforts to implement these abstract models in more detailed biophysical compartmental simulations raise specific issues that could be addressed in larger scale population models incorporating mechanisms of inhibition.

Functional phase response curves: a method for understanding synchronization of adapting neurons

Phase response curves (PRCs) for a single neuron are often used to predict the synchrony of mutually coupled neurons. Previous theoretical work on pulse-coupled oscillators used single-pulse perturbations. We propose an alternate method in which functional PRCs (fPRCs) are generated using a train of pulses applied at a fixed delay after each spike, with the PRC measured when the phasic relationship between the stimulus and the subsequent spike in the neuron has converged. The essential information is the dependence of the recovery time from pulse onset until the next spike as a function of the delay between the previous spike and the onset of the applied pulse. Experimental fPRCs in Aplysia pacemaker neurons were different from single-pulse PRCs, principally due to adaptation. In the biological neuron, convergence to the fully adapted recovery interval was slower at some phases than that at others because the change in the effective intrinsic period due to adaptation changes the effective phase resetting in a way that opposes and slows the effects of adaptation. The fPRCs for two isolated adapting model neurons were used to predict the existence and stability of 1:1 phase-locked network activity when the two neurons were coupled. A stability criterion was derived by linearizing a coupled map based on the fPRC and the existence and stability criteria were successfully tested in two-simulated-neuron networks with reciprocal inhibition or excitation. The fPRC is the first PRC-based tool that can account for adaptation in analyzing networks of neural oscillators.

Phase-resetting curves determine synchronization, phase locking, and clustering in networks of neural oscillators

Networks of model neurons were constructed and their activity was predicted using an iterated map based solely on the phase-resetting curves (PRCs). The predictions were quite accurate provided that the resetting to simultaneous inputs was calculated using the sum of the simultaneously active conductances, obviating the need for weak coupling assumptions. Fully synchronous activity was observed only when the slope of the PRC at a phase of zero, corresponding to spike initiation, was positive. A novel stability criterion was developed and tested for all-to-all networks of identical, identically connected neurons. When the PRC generated using N-1 simultaneously active inputs becomes too steep, the fully synchronous mode loses stability in a network of N model neurons. Therefore, the stability of synchrony can be lost by increasing the slope of this PRC either by increasing the network size or the strength of the individual synapses. Existence and stability criteria were also developed and tested for the splay mode in which neurons fire sequentially. Finally, N/M synchronous subclusters of M neurons were predicted using the intersection of parameters that supported both between-cluster splay and within-cluster synchrony. Surprisingly, the splay mode between clusters could enforce synchrony on subclusters that were incapable of synchronizing themselves. These results can be used to gain insights into the activity of networks of biological neurons whose PRCs can be measured.

Potential network mechanisms mediating electroencephalographic beta rhythm changes during propofol-induced paradoxical excitation

Propofol, like most general anesthetic drugs, can induce both behavioral and electroencephalographic (EEG) manifestations of excitation, rather than sedation, at low doses. Neuronal excitation is unexpected in the presence of this GABA(A)-potentiating drug. We construct a series of network models to understand this paradox. Individual neurons have ion channel conductances with Hodgkin-Huxley-type formulations. Propofol increases the maximal conductance and time constant of decay of the synaptic GABA(A) current. Networks range in size from 2 to 230 neurons. Population output is measured as a function of pyramidal cell activity, with the electroencephalogram approximated by the sum of population AMPA activity between pyramidal cells. These model networks suggest propofol-induced paradoxical excitation may result from a membrane level interaction between the GABA(A) current and an intrinsic membrane slow potassium current (M-current). This membrane level interaction has consequences at the level of multicellular networks enabling a switch from baseline interneuron synchrony to propofol-induced interneuron antisynchrony. Large network models reproduce the clinical EEG changes characteristic of propofol-induced paradoxical excitation. The EEG changes coincide with the emergence of antisynchronous interneuron clusters in the model networks. Our findings suggest interneuron antisynchrony as a potential network mechanism underlying the generation of propofol-induced paradoxical excitation. As correlates of behavioral phenomenology, these networks may refine our understanding of the specific behavioral states associated with general anesthesia.

Using phase resetting to predict 1:1 and 2:2 locking in two neuron networks in which firing order is not always preserved

Our goal is to understand how nearly synchronous modes arise in heterogenous networks of neurons. In heterogenous networks, instead of exact synchrony, nearly synchronous modes arise, which include both 1:1 and 2:2 phase-locked modes. Existence and stability criteria for 2:2 phase-locked modes in reciprocally coupled two neuron circuits were derived based on the open loop phase resetting curve (PRC) without the assumption of weak coupling. The PRC for each component neuron was generated using the change in synaptic conductance produced by a presynaptic action potential as the perturbation. Separate derivations were required for modes in which the firing order is preserved and for those in which it alternates. Networks composed of two model neurons coupled by reciprocal inhibition were examined to test the predictions. The parameter regimes in which both types of nearly synchronous modes are exhibited were accurately predicted both qualitatively and quantitatively provided that the synaptic time constant is short with respect to the period and that the effect of second order resetting is considered. In contrast, PRC methods based on weak coupling could not predict 2:2 modes and did not predict the 1:1 modes with the level of accuracy achieved by the strong coupling methods. The strong coupling prediction methods provide insight into what manipulations promote near-synchrony in a two neuron network and may also have predictive value for larger networks, which can also manifest changes in firing order. We also identify a novel route by which synchrony is lost in mildly heterogenous networks.

Towards blueprints for network architecture, biophysical dynamics and signal transduction

We review mathematical aspects of biophysical dynamics, signal transduction and network architecture that have been used to uncover functionally significant relations between the dynamics of single neurons and the networks they compose. We focus on examples that combine insights from these three areas to expand our understanding of systems neuroscience. These range from single neuron coding to models of decision making and electrosensory discrimination by networks and populations and also coincidence detection in pairs of dendrites and dynamics of large networks of excitable dendritic spines. We conclude by describing some of the challenges that lie ahead as the applied mathematics community seeks to provide the tools which will ultimately underpin systems neuroscience.

The dynamic structure underlying subthreshold oscillatory activity and the onset of spikes in a model of medial entorhinal cortex stellate cells

Medial entorhinal cortex layer II stellate cells display subthreshold oscillations (STOs). We study a single compartment biophysical model of such cells which qualitatively reproduces these STOs. We argue that in the subthreshold interval (STI) the seven-dimensional model can be reduced to a three-dimensional system of equations with well differentiated times scales. Using dynamical systems arguments we provide a mechanism for generations of STOs. This mechanism is based on the "canard structure," in which relevant trajectories stay close to repelling manifolds for a significant interval of time. We also show that the transition from subthreshold oscillatory activity to spiking ("canard explosion") is controlled in the STI by the same structure. A similar mechanism is invoked to explain why noise increases the robustness of the STO regime. Taking advantage of the reduction of the dimensionality of the full stellate cell system, we propose a nonlinear artificially spiking (NAS) model in which the STI reduced system is supplemented with a threshold for spiking and a reset voltage. We show that the synchronization properties in networks made up of the NAS cells are similar to those of networks using the full stellate cell models.