Coincidence detection in the Hodgkin-Huxley equations

Some of the cochlear nuclei in the auditory pathway are specialized for the sound localization. They compute the interaural time difference. The difference in sound timing is transduced by the dedicated neuronal circuit into a labeled line difference. The detector neurons along the delay line fire only when synaptic inputs reflecting signals from both cars arrive within a short time window. It was therefore called coincidence detection. We show, (1) what are the limits of coincidence detection in the leaky integrator model, which is a linear system, (2) how should the ideal coincidence detector based on the Hodkin-Huxley equations from real neurons look like, (3) what are the properties and physical limits in the real coincidence detection system. The conclusion is that the neuron with the Hodgkin Huxley dynamics has a fixed precision for the coincidence detection. The limits of the sound localization precision are set by the frequency of the sound and, therefore, by the vector strength of spike trains generated in the neuronal circuit in response to the sound.     

Recurrent Network Models for Perfect Temporal Integration of Fluctuating Correlated Inputs

Temporal integration of input is essential to the accumulation of information in various cognitive and behavioral processes, and gradually increasing neuronal activity, typically occurring within a range of seconds, is considered to reflect such computation by the brain. Some psychological evidence suggests that temporal integration by the brain is nearly perfect, that is, the integration is non-leaky, and the output of a neural integrator is accurately proportional to the strength of input. Neural mechanisms of perfect temporal integration, however, remain largely unknown. Here, we propose a recurrent network model of cortical neurons that perfectly integrates partially correlated, irregular input spike trains. We demonstrate that the rate of this temporal integration changes proportionately to the probability of spike coincidences in synaptic inputs. We analytically prove that this highly accurate integration of synaptic inputs emerges from integration of the variance of the fluctuating synaptic inputs, when their mean component is kept constant. Highly irregular neuronal firing and spike coincidences are the major features of cortical activity, but they have been separately addressed so far. Our results suggest that the efficient protocol of information integration by cortical networks essentially requires both features and hence is heterotic.     

Maturation of GABAergic Inhibition Promotes Strengthening of Temporally Coherent Inputs among Convergent Pathways

Spike-timing-dependent plasticity (STDP), a form of Hebbian plasticity, is inherently stabilizing. Whether and how GABAergic inhibition influences STDP is not well understood. Using a model neuron driven by converging inputs modifiable by STDP, we determined that a sufficient level of inhibition was critical to ensure that temporal coherence (correlation among presynaptic spike times) of synaptic inputs, rather than initial strength or number of inputs within a pathway, controlled postsynaptic spike timing. Inhibition exerted this effect by preferentially reducing synaptic efficacy, the ability of inputs to evoke postsynaptic action potentials, of the less coherent inputs. In visual cortical slices, inhibition potently reduced synaptic efficacy at ages during but not before the critical period of ocular dominance (OD) plasticity. Whole-cell recordings revealed that the amplitude of unitary IPSCs from parvalbumin positive (Pv+) interneurons to pyramidal neurons increased during the critical period, while the synaptic decay time-constant decreased. In addition, intrinsic properties of Pv+ interneurons matured, resulting in an increase in instantaneous firing rate. Our results suggest that maturation of inhibition in visual cortex ensures that the temporally coherent inputs (e.g. those from the open eye during monocular deprivation) control postsynaptic spike times of binocular neurons, a prerequisite for Hebbian mechanisms to induce OD plasticity.     

Identifying and Tracking Simulated Synaptic Inputs from Neuronal Firing: Insights from In Vitro Experiments

Accurately describing synaptic interactions between neurons and how interactions change over time are key challenges for systems neuroscience. Although intracellular electrophysiology is a powerful tool for studying synaptic integration and plasticity, it is limited by the small number of neurons that can be recorded simultaneously in vitro and by the technical difficulty of intracellular recording in vivo. One way around these difficulties may be to use large-scale extracellular recording of spike trains and apply statistical methods to model and infer functional connections between neurons. These techniques have the potential to reveal large-scale connectivity structure based on the spike timing alone. However, the interpretation of functional connectivity is often approximate, since only a small fraction of presynaptic inputs are typically observed. Here we use in vitro current injection in layer 2/3 pyramidal neurons to validate methods for inferring functional connectivity in a setting where input to the neuron is controlled. In experiments with partially-defined input, we inject a single simulated input with known amplitude on a background of fluctuating noise. In a fully-defined input paradigm, we then control the synaptic weights and timing of many simulated presynaptic neurons. By analyzing the firing of neurons in response to these artificial inputs, we ask 1) How does functional connectivity inferred from spikes relate to simulated synaptic input? and 2) What are the limitations of connectivity inference? We find that individual current-based synaptic inputs are detectable over a broad range of amplitudes and conditions. Detectability depends on input amplitude and output firing rate, and excitatory inputs are detected more readily than inhibitory. Moreover, as we model increasing numbers of presynaptic inputs, we are able to estimate connection strengths more accurately and detect the presence of connections more quickly. These results illustrate the possibilities and outline the limits of inferring synaptic input from spikes.     

Enhancement of spike-timing precision by autaptic transmission in neocortical inhibitory interneurons

In vivo studies suggest that precise firing of neurons is important for correct sensory representation. Principal neocortical neurons fire imprecisely when repeatedly activated by fixed sensory stimuli or current depolarizations. Here we show that in contrast to pyramidal neurons, firing in neocortical GABAergic fast-spiking (FS) interneurons is quite precise. FS interneurons are self-innervated by powerful GABAergic autaptic connections reliably activated after each spike, suggesting that autapses strongly regulate FS-cell spike timing. Indeed, blockade of autaptic transmission degraded temporal precision in multiple ways. Under these conditions, realistic dynamic-clamp hyperpolarizing autapses restored precision of spike timing, even in the presence of synaptic noise. Furthermore, firing precision was increased in pyramidal neurons by artificial GABAergic autaptic conductances, suggesting that tightly coupled synaptic feedback inhibition regulates spike timing in principal cells. Thus, well-timed inhibition, whether autaptic or synaptic, facilitates precise spike timing and promotes synchronized cortical network oscillations relevant to several behaviors.     

Thematic Minireview Series: Molecular Mechanisms of Synaptic Plasticity

The human brain contains ∼86 billion neurons, which are precisely organized in specific brain regions and nuclei. High fidelity synaptic communication between subsets of neurons in specific circuits is required for most human behaviors, and is often disrupted in neuropsychiatric disorders. The presynaptic axon terminals of one neuron release neurotransmitters that activate receptors on multiple postsynaptic neuron targets to induce electrical and chemical responses. Typically, postsynaptic neurons integrate signals from multiple presynaptic neurons at thousands of synaptic inputs to control downstream communication to the next neuron in the circuit. Importantly, the strength (or efficiency) of signal transmission at each synapse can be modulated on time scales ranging up to the lifetime of the organism. This “synaptic plasticity” leads to changes in overall neuronal circuit activity, resulting in behavioral modifications. This series of minireviews will focus on recent advances in our understanding of the molecular and cellular mechanisms that control synaptic plasticity.     

The time window for generation of dendritic spikes by coincidence of action potentials and EPSPs is layer specific in somatosensory cortex

The precise timing of events in the brain has consequences for intracellular processes, synaptic plasticity, integration and network behaviour. Pyramidal neurons, the most widespread excitatory neuron of the neocortex have multiple spike initiation zones, which interact via dendritic and somatic spikes actively propagating in all directions within the dendritic tree. For these neurons, therefore, both the location and timing of synaptic inputs are critical. The time window for which the backpropagating action potential can influence dendritic spike generation has been extensively studied in layer 5 neocortical pyramidal neurons of rat somatosensory cortex. Here, we re-examine this coincidence detection window for pyramidal cell types across the rat somatosensory cortex in layers 2/3, 5 and 6. We find that the time-window for optimal interaction is widest and shifted in layer 5 pyramidal neurons relative to cells in layers 6 and 2/3. Inputs arriving at the same time and locations will therefore differentially affect spike-timing dependent processes in the different classes of pyramidal neurons.     

Variability and coding efficiency of noisy neural spike encoders

Encoding synaptic inputs as a train of action potentials is a fundamental function of nerve cells. Although spike trains recorded in vivo have been shown to be highly variable, it is unclear whether variability in spike timing represents faithful encoding of temporally varying synaptic inputs or noise inherent in the spike encoding mechanism. It has been reported that spike timing variability is more pronounced for constant, unvarying inputs than for inputs with rich temporal structure. This could have significant implications for the nature of neural coding, particularly if precise timing of spikes and temporal synchrony between neurons is used to represent information in the nervous system. To study the potential functional role of spike timing variability, we estimate the fraction of spike timing variability which conveys information about the input for two types of noisy spike encoders--an integrate and fire model with randomly chosen thresholds and a model of a patch of neuronal membrane containing stochastic Na(+) and K(+) channels obeying Hodgkin-Huxley kinetics. The quality of signal encoding is assessed by reconstructing the input stimuli from the output spike trains using optimal linear mean square estimation. A comparison of the estimation performance of noisy neuronal models of spike generation enables us to assess the impact of neuronal noise on the efficacy of neural coding. The results for both models suggest that spike timing variability reduces the ability of spike trains to encode rapid time-varying stimuli. Moreover, contrary to expectations based on earlier studies, we find that the noisy spike encoding models encode slowly varying stimuli more effectively than rapidly varying ones.     

Stochastically Gating Ion Channels Enable Patterned Spike Firing through Activity-Dependent Modulation of Spike Probability

The transformation of synaptic input into patterns of spike output is a fundamental operation that is determined by the particular complement of ion channels that a neuron expresses. Although it is well established that individual ion channel proteins make stochastic transitions between conducting and non-conducting states, most models of synaptic integration are deterministic, and relatively little is known about the functional consequences of interactions between stochastically gating ion channels. Here, we show that a model of stellate neurons from layer II of the medial entorhinal cortex implemented with either stochastic or deterministically gating ion channels can reproduce the resting membrane properties of stellate neurons, but only the stochastic version of the model can fully account for perithreshold membrane potential fluctuations and clustered patterns of spike output that are recorded from stellate neurons during depolarized states. We demonstrate that the stochastic model implements an example of a general mechanism for patterning of neuronal output through activity-dependent changes in the probability of spike firing. Unlike deterministic mechanisms that generate spike patterns through slow changes in the state of model parameters, this general stochastic mechanism does not require retention of information beyond the duration of a single spike and its associated afterhyperpolarization. Instead, clustered patterns of spikes emerge in the stochastic model of stellate neurons as a result of a transient increase in firing probability driven by activation of HCN channels during recovery from the spike afterhyperpolarization. Using this model, we infer conditions in which stochastic ion channel gating may influence firing patterns in vivo and predict consequences of modifications of HCN channel function for in vivo firing patterns.     

Optimal stimulus shapes for neuronal excitation

An important problem in neuronal computation is to discern how features of stimuli control the timing of action potentials. One aspect of this problem is to determine how an action potential, or spike, can be elicited with the least energy cost, e.g., a minimal amount of applied current. Here we show in the Hodgkin & Huxley model of the action potential and in experiments on squid giant axons that: 1) spike generation in a neuron can be highly discriminatory for stimulus shape and 2) the optimal stimulus shape is dependent upon inputs to the neuron. We show how polarity and time course of post-synaptic currents determine which of these optimal stimulus shapes best excites the neuron. These results are obtained mathematically using the calculus of variations and experimentally using a stochastic search methodology. Our findings reveal a surprising complexity of computation at the single cell level that may be relevant for understanding optimization of signaling in neurons and neuronal networks.     

Synchronization and sensitivity enhancement of the Hodgkin-Huxley neurons due to inhibitory inputs

Recent experimental results imply that inhibitory postsynaptic potentials can play a functional role in realizing synchronization of neuronal firing in the brain. In order to examine the relation between inhibition and synchronous firing of neurons theoretically, we analyze possible effects of synchronization and sensitivity enhancement caused by inhibitory inputs to neurons with a biologically realistic model of the Hodgkin-Huxley equations. The result shows that, after an inhibitory spike, the firing probability of a single postsynaptic neuron exposed to random excitatory background activity oscillates with time. The oscillation of the firing probability can be related to synchronous firing of neurons receiving an inhibitory spike simultaneously. Further, we show that when an inhibitory spike input precedes an excitatory spike input, the presence of such preceding inhibition raises the firing probability peak of the neuron after the excitatory input. The result indicates that an inhibitory spike input can enhance the sensitivity of the postsynaptic neuron to the following excitatory spike input. Two neural network models based on these effects on postsynaptic neurons caused by inhibitory inputs are proposed to demonstrate possible mechanisms of detecting particular spatiotemporal spike patterns. Received: 15 April 1999 /Accepted in revised form: 25 November 1999     

Afterhyperpolarization improves spike programming through lowering threshold potentials and refractory periods mediated by voltage-gated sodium channels

Neurons program various patterns of sequential spikes as neural codes to guide animal behavior. Studies show that spike programming (capacity and timing precision) is influenced by inhibitory synaptic inputs and membrane afterhyperpolarization (AHP). Less is clear about how these inhibitory components regulate spike programming, which we investigated at the cortical neurons. Whole-cell current-clamp recording for action potentials and single channel recording for voltage-gated sodium channels (VGSC) were conducted at regular-spiking and fast-spiking neurons in the cortical slices. With quantifying the threshold potentials and refractory periods of sequential spikes, we found that fast-spiking neurons expressing AHP possess lower threshold potentials and shorter refractory periods, and the hyperpolarization pulse immediately after each of spikes lowers threshold potentials and shortens refractory periods at regular-spiking neurons. Moreover, the hyperpolarization pulses shorten the refractory periods for VGSC reactivation and threshold potentials for its sequential activation. Our data indicate that inhibitory components immediately after spikes, such as AHP and recurrent inhibition, improve spike capacity and timing precision via lowering the refractory periods and threshold potentials mediated by voltage-gated sodium channels.     

Spontaneously emerging direction selectivity maps in visual cortex through STDP

It is still an open question as to whether, and how, direction-selective neuronal responses in primary visual cortex are generated by feedforward thalamocortical or recurrent intracortical connections, or a combination of both. Here we present an investigation that concentrates on and, only for the sake of simplicity, restricts itself to intracortical circuits, in particular, with respect to the developmental aspects of direction selectivity through spike-timing-dependent synaptic plasticity. We show that directional responses can emerge in a recurrent network model of visual cortex with spiking neurons that integrate inputs mainly from a particular direction, thus giving rise to an asymmetrically shaped receptive field. A moving stimulus that enters the receptive field from this (preferred) direction will activate a neuron most strongly because of the increased number and/or strength of inputs from this direction and since delayed isotropic inhibition will neither overlap with, nor cancel excitation, as would be the case for other stimulus directions. It is demonstrated how direction-selective responses result from spatial asymmetries in the distribution of synaptic contacts or weights of inputs delivered to a neuron by slowly conducting intracortical axonal delay lines. By means of spike-timing-dependent synaptic plasticity with an asymmetric learning window this kind of coupling asymmetry develops naturally in a recurrent network of stochastically spiking neurons in a scenario where the neurons are activated by unidirectionally moving bar stimuli and even when only intrinsic spontaneous activity drives the learning process. We also present simulation results to show the ability of this model to produce direction preference maps similar to experimental findings     

Structure-preserving model reduction of passive and quasi-active neurons

The spatial component of input signals often carries information crucial to a neuron’s function, but models mapping synaptic inputs to the transmembrane potential can be computationally expensive. Existing reduced models of the neuron either merge compartments, thereby sacrificing the spatial specificity of inputs, or apply model reduction techniques that sacrifice the underlying electrophysiology of the model. We use Krylov subspace projection methods to construct reduced models of passive and quasi-active neurons that preserve both the spatial specificity of inputs and the electrophysiological interpretation as an RC and RLC circuit, respectively. Each reduced model accurately computes the potential at the spike initiation zone (SIZ) given a much smaller dimension and simulation time, as we show numerically and theoretically. The structure is preserved through the similarity in the circuit representations, for which we provide circuit diagrams and mathematical expressions for the circuit elements. Furthermore, the transformation from the full to the reduced system is straightforward and depends on intrinsic properties of the dendrite. As each reduced model is accurate and has a clear electrophysiological interpretation, the reduced models can be used not only to simulate morphologically accurate neurons but also to examine computations performed in dendrites.     

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.     

Dendritic I h Selectively Blocks Temporal Summation of Unsynchronized Distal Inputs in CA1 Pyramidal Neurons

The active dendritic conductances shape the input-output properties of many principal neurons in different brain regions, and the various ways in which they regulate neuronal excitability need to be investigated to better understand their functional consequences. Using a realistic model of a hippocampal CA1 pyramidal neuron, we show a major role for the hyperpolarization-activated current, Ih, in regulating the spike probability of a neuron when independent synaptic inputs are activated with different degrees of synchronization and at different distances from the soma. The results allowed us to make the experimentally testable prediction that the Ih in these neurons is needed to reduce neuronal excitability selectively for distal unsynchronized, but not for synchronized, inputs.     

Responses of a bursting pacemaker to excitation reveal spatial segregation between bursting and spiking mechanisms

Central pattern generators (CPGs) frequently include bursting neurons that serve as pacemakers for rhythm generation. Phase resetting curves (PRCs) can provide insight into mechanisms underlying phase locking in such circuits. PRCs were constructed for a pacemaker bursting complex in the pyloric circuit in the stomatogastric ganglion of the lobster and crab. This complex is comprised of the Anterior Burster (AB) neuron and two Pyloric Dilator (PD) neurons that are all electrically coupled. Artificial excitatory synaptic conductance pulses of different strengths and durations were injected into one of the AB or PD somata using the Dynamic Clamp. Previously, we characterized the inhibitory PRCs by assuming a single slow process that enabled synaptic inputs to trigger switches between an up state in which spiking occurs and a down state in which it does not. Excitation produced five different PRC shapes, which could not be explained with such a simple model. A separate dendritic compartment was required to separate the mechanism that generates the up and down phases of the bursting envelope (1) from synaptic inputs applied at the soma, (2) from axonal spike generation and (3) from a slow process with a slower time scale than burst generation. This study reveals that due to the nonlinear properties and compartmentalization of ionic channels, the response to excitation is more complex than inhibition.     

The response of cortical neurons to in vivo-like input current: theory and experiment: II. Time-varying and spatially distributed inputs

The response of a population of neurons to time-varying synaptic inputs can show a rich phenomenology, hardly predictable from the dynamical properties of the membrane’s inherent time constants. For example, a network of neurons in a state of spontaneous activity can respond significantly more rapidly than each single neuron taken individually. Under the assumption that the statistics of the synaptic input is the same for a population of similarly behaving neurons (mean field approximation), it is possible to greatly simplify the study of neural circuits, both in the case in which the statistics of the input are stationary (reviewed in La Camera et al. in Biol Cybern, 2008) and in the case in which they are time varying and unevenly distributed over the dendritic tree. Here, we review theoretical and experimental results on the single-neuron properties that are relevant for the dynamical collective behavior of a population of neurons. We focus on the response of integrate-and-fire neurons and real cortical neurons to long-lasting, noisy, in vivo-like stationary inputs and show how the theory can predict the observed rhythmic activity of cultures of neurons. We then show how cortical neurons adapt on multiple time scales in response to input with stationary statistics in vitro. Next, we review how it is possible to study the general response properties of a neural circuit to time-varying inputs by estimating the response of single neurons to noisy sinusoidal currents. Finally, we address the dendrite–soma interactions in cortical neurons leading to gain modulation and spike bursts, and show how these effects can be captured by a two-compartment integrate-and-fire neuron. Most of the experimental results reviewed in this article have been successfully reproduced by simple integrate-and-fire model neurons.     

Membrane Potential Fluctuations Determine the Precision of Spike Timing and Synchronous Activity: A Model Study

It is much debated on what time scale information is encoded by neuronal spike activity. With a phenomenological model that transforms time-dependent membrane potential fluctuations into spike trains, we investigate constraints for the timing of spikes and for synchronous activity of neurons with common input. The model of spike generation has a variable threshold that depends on the time elapsed since the previous action potential and on the preceding membrane potential changes. To ensure that the model operates in a biologically meaningful range, the model was adjusted to fit the responses of a fly visual interneuron to motion stimuli. The dependence of spike timing on the membrane potential dynamics was analyzed. Fast membrane potential fluctuations are needed to trigger spikes with a high temporal precision. Slow fluctuations lead to spike activity with a rate about proportional to the membrane potential. Thus, for a given level of stochastic input, the frequency range of membrane potential fluctuations induced by a stimulus determines whether a neuron can use a rate code or a temporal code. The relationship between the steepness of membrane potential fluctuations and the timing of spikes has also implications for synchronous activity in neurons with common input. Fast membrane potential changes must be shared by the neurons to produce synchronous activity.