Model of gamma frequency burst discharge generated by conditional backpropagation

Impact on backpropagation of the spatial heterogeneity of sodium channel kinetics in the axon initial segment

In a variety of neurons, action potentials (APs) initiate at the proximal axon, within a region called the axon initial segment (AIS), which has a high density of voltage-gated sodium channels (NaVs) on its membrane. In pyramidal neurons, the proximal AIS has been reported to exhibit a higher proportion of NaVs with gating properties that are "right-shifted" to more depolarized voltages, compared to the distal AIS. Further, recent experiments have revealed that as neurons develop, the spatial distribution of NaV subtypes along the AIS can change substantially, suggesting that neurons tune their excitability by modifying said distribution. When neurons are stimulated axonally, computational modelling has shown that this spatial separation of gating properties in the AIS enhances the backpropagation of APs into the dendrites. In contrast, in the more natural scenario of somatic stimulation, our simulations show that the same distribution can impede backpropagation, suggesting that the choice of orthodromic versus antidromic stimulation can bias or even invert experimental findings regarding the role of NaV subtypes in the AIS. We implemented a range of hypothetical NaV distributions in the AIS of three multicompartmental pyramidal cell models and investigated the precise kinetic mechanisms underlying such effects, as the spatial distribution of NaV subtypes is varied. With axonal stimulation, proximal NaV availability dominates, such that concentrating right-shifted NaVs in the proximal AIS promotes backpropagation. However, with somatic stimulation, the models are insensitive to availability kinetics. Instead, the higher activation threshold of right-shifted NaVs in the AIS impedes backpropagation. Therefore, recently observed developmental changes to the spatial separation and relative proportions of NaV1.2 and NaV1.6 in the AIS differentially impact activation and availability. The observed effects on backpropagation, and potentially learning via its putative role in synaptic plasticity (e.g. through spike-timing-dependent plasticity), are opposite for orthodromic versus antidromic stimulation, which should inform hypotheses about the impact of the developmentally regulated subcellular localization of these NaV subtypes.

Do Biological Constraints Impair Dendritic Computation?

Computations on the dendritic trees of neurons have important constraints. Voltage dependent conductances in dendrites are not similar to arbitrary direct-current generation, they are the basis for dendritic nonlinearities and they do not allow converting positive currents into negative currents. While it has been speculated that the dendritic tree of a neuron can be seen as a multi-layer neural network and it has been shown that such an architecture could be computationally strong, we do not know if that computational strength is preserved under these biological constraints. Here we simulate models of dendritic computation with and without these constraints. We find that dendritic model performance on interesting machine learning tasks is not hurt by these constraints but may benefit from them. Our results suggest that single real dendritic trees may be able to learn a surprisingly broad range of tasks.

Classes of dendritic information processing

Dendrites are much more than passive neuronal components. Mounting experimental evidence and decades of computational work have decisively shown that dendrites leverage a host of nonlinear biophysical phenomena and actively participate in sophisticated computations, at the level of the single neuron and at the level of the network. However, a coherent view of their processing power is still lacking and dendrites are largely neglected in neural network models. Here, we describe four classes of dendritic information processing and delineate their implications at the algorithmic level. We propose that beyond the well-known spatiotemporal filtering of their inputs, dendrites are capable of selecting, routing and multiplexing information. By separating dendritic processing from axonal outputs, neuron networks gain a degree of freedom with implications for perception and learning.

Burst Firing in the Electrosensory System of Gymnotiform Weakly Electric Fish: Mechanisms and Functional Roles

Neurons across sensory systems and organisms often display complex patterns of action potentials in response to sensory input. One example of such a pattern is the tendency of neurons to fire packets of action potentials (i.e., a burst) followed by quiescence. While it is well known that multiple mechanisms can generate bursts of action potentials at both the single-neuron and the network level, the functional role of burst firing in sensory processing is not so well understood to date. Here we provide a comprehensive review of the known mechanisms and functions of burst firing in processing of electrosensory stimuli in gymnotiform weakly electric fish. We also present new evidence from existing data showing that bursts and isolated spikes provide distinct information about stimulus variance. It is likely that these functional roles will be generally applicable to other systems and species.

Learning contrast-invariant cancellation of redundant signals in neural systems

Cancellation of redundant information is a highly desirable feature of sensory systems, since it would potentially lead to a more efficient detection of novel information. However, biologically plausible mechanisms responsible for such selective cancellation, and especially those robust to realistic variations in the intensity of the redundant signals, are mostly unknown. In this work, we study, via in vivo experimental recordings and computational models, the behavior of a cerebellar-like circuit in the weakly electric fish which is known to perform cancellation of redundant stimuli. We experimentally observe contrast invariance in the cancellation of spatially and temporally redundant stimuli in such a system. Our model, which incorporates heterogeneously-delayed feedback, bursting dynamics and burst-induced STDP, is in agreement with our in vivo observations. In addition, the model gives insight on the activity of granule cells and parallel fibers involved in the feedback pathway, and provides a strong prediction on the parallel fiber potentiation time scale. Finally, our model predicts the existence of an optimal learning contrast around 15% contrast levels, which are commonly experienced by interacting fish.

Modeling cancelation of periodic inputs with burst-STDP and feedback

Prediction and cancelation of redundant information is an important feature that many neural systems must display in order to efficiently code external signals. We develop an analytic framework for such cancelation in sensory neurons produced by a cerebellar-like structure in wave-type electric fish. Our biologically plausible mechanism is motivated by experimental evidence of cancelation of periodic input arising from the proximity of conspecifics as well as tail motion. This mechanism involves elements present in a wide range of systems: (1) stimulus-driven feedback to the neurons acting as detectors, (2) a large variety of temporal delays in the pathways transmitting such feedback, responsible for producing frequency channels, and (3) burst-induced long-term plasticity. The bursting arises from back-propagating action potentials. Bursting events drive the input frequency-dependent learning rule, which in turn affects the feedback input and thus the burst rate. We show how the mean firing rate and the rate of production of 2- and 4-spike bursts (the main learning events) can be estimated analytically for a leaky integrate-and-fire model driven by (slow) sinusoidal, back-propagating and feedback inputs as well as rectified filtered noise. The effect of bursts on the average synaptic strength is also derived. Our results shed light on why bursts rather than single spikes can drive learning in such networks "online", i.e. in the absence of a correlative discharge. Phase locked spiking in frequency specific channels together with a frequency-dependent STDP window size regulate burst probability and duration self-consistently to implement cancelation.

Frequency-tuned cerebellar channels and burst-induced LTD lead to the cancellation of redundant sensory inputs

For optimal sensory processing, neural circuits must extract novel, unpredictable signals from the redundant sensory input in which they are embedded, but the detailed cellular and network mechanisms that implement such selective cancellation are presently unknown. Using a combination of modeling and experiment, we characterize in detail a cerebellar circuit in weakly electric fish, showing how it can carry out this computation. We use a model incorporating the wide range of experimentally estimated parallel fiber feedback delays and a burst-induced LTD rule derived from in vitro experiments to explain the precise cancellation of redundant signals observed in vivo. Our model demonstrates how the backpropagation-dependent burst dynamics adjusts the temporal pairing width of the plasticity mechanism to precisely match the frequency of the redundant signal. The model also makes the prediction that this cerebellar feedback pathway must be composed of frequency-tuned channels; this prediction is subsequently verified in vivo, highlighting a novel and general capability of cerebellar circuitry.

Routing the flow of sensory signals using plastic responses to bursts and isolated spikes: experiment and theory

Processing complex sensory environments efficiently requires a diverse array of neural coding strategies. Neural codes relying on specific temporal patterning of action potentials may offer advantages over using solely spike rate codes. In particular, stimulus-dependent burst firing may carry additional information that isolated spikes do not. We use the well characterized electrosensory system of weakly electric fish to address how stimulus-dependent burst firing can determine the flow of information in feedforward neural circuits with different forms of short-term synaptic plasticity. Pyramidal cells in the electrosensory lateral line lobe burst in response to low-frequency, local (prey) signals. We show that the ability of pyramidal cells to code for local signals in the presence of additional high-frequency, global (communication) stimuli is uncompromised, while burst firing is reduced. We developed a bursting neuron model to understand how these effects, in particular noise-induced burst suppression, arise from interplay between incoming sensory signals and intrinsic neuronal dynamics. Finally, we examined how postsynaptic target populations preferentially respond to one of the two sensory mixtures (local vs local plus global) depending on whether the populations are in receipt of facilitating or depressing synapses. This form of feedforward neural architecture may allow for efficient information flow in the same neural pathway via either isolated or burst spikes, where the mechanisms by which stimuli are encoded are adaptable and sensitive to a diverse array of stimulus and contextual mixtures.

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.

Neural heterogeneities and stimulus properties affect burst coding in vivo

Many neurons tend to fire clusters of action potentials called bursts followed by quiescence in response to sensory input. While the mechanisms that underlie burst firing are generally well understood in vitro, the functional role of these bursts in generating behavioral responses to sensory input in vivo are less clear. Pyramidal cells within the electrosensory lateral line lobe (ELL) of weakly electric fish offer an attractive model system for studying the coding properties of burst firing, because the anatomy and physiology of the electrosensory circuitry are well understood, and the burst mechanism of ELL pyramidal cells has been thoroughly characterized in vitro. We investigated the coding properties of bursts generated by these cells in vivo in response to mimics of behaviorally relevant sensory input. We found that heterogeneities within the pyramidal cell population had quantitative but not qualitative effects on burst coding for the low frequency components of broadband time varying input. Moreover, spatially localized stimuli mimicking, for example, prey tended to elicit more bursts than spatially global stimuli mimicking conspecific-related stimuli. We also found small but significant correlations between burst attributes such as the number of spikes per burst or the interspike interval during the burst and stimulus attributes such as stimulus amplitude or slope. These correlations were much weaker in magnitude than those observed in vitro. More surprisingly, our results show that correlations between burst and stimulus attributes actually decreased in magnitude when we used low frequency stimuli that are expected to promote burst firing. We propose that this discrepancy is attributable to differences between ELL pyramidal cell burst firing under in vivo and in vitro conditions.

SK channels gate information processing in vivo by regulating an intrinsic bursting mechanism seen in vitro

Understanding the mechanistic substrates of neural computations that lead to behavior remains a fundamental problem in neuroscience. In particular, the contributions of intrinsic neural properties such as burst firing and dendritic morphology to the processing of behaviorally relevant sensory input have received much interest recently. Pyramidal cells within the electrosensory lateral line lobe of weakly electric fish display an intrinsic bursting mechanism that relies on somato-dendritic interactions when recorded in vitro: backpropagating somatic action potentials trigger dendritic action potentials that lead to a depolarizing afterpotential (DAP) at the soma. We recorded intracellularly from these neurons in vivo and found firing patterns that were quite different from those seen in vitro: we found no evidence for DAPs as each somatic action potential was followed by a pronounced afterhyperpolarization (AHP). Calcium chelators injected in vivo reduced the AHP, thereby unmasking the DAP and inducing in vitro-like bursting in pyramidal cells. These bursting dynamics significantly reduced the cell's ability to encode the detailed time course of sensory input. We performed additional in vivo pharmacological manipulations and mathematical modeling to show that calcium influx through N-methyl-d-aspartate (NMDA) receptors activate dendritic small conductance (SK) calcium-activated potassium channels, which causes an AHP that counteracts the DAP and leads to early termination of the burst. Our results show that ion channels located in dendrites can have a profound influence on the processing of sensory input by neurons in vivo through the modulation of an intrinsic bursting mechanism.

Regulation of somatic firing dynamics by backpropagating dendritic spikes

Pyramidal cells of the apteronotid ELL have been shown to display a characteristic mechanism of burst discharge, which has been shown to play an important role in sensory coding. This form of bursting depends on a reciprocal dendro-somatic interaction, in which discharge of a somatic spike causes a dendritic spike, which in turn contributes a dendro-somatic current flow to create a depolarizing afterpotential (DAP) in the soma. We review here our recent work showing how the timing of this DAP influences the somatic firing dynamics, and how the degree of inactivation of dendritic Na(+) currents can cause an increased delay between somatic and dendritic spikes. This ultimately allows the DAP to become more effective at increasing the excitability of the somatic spike generating mechanism. Further, this delay between dendritic and somatic spiking can be regulated by strongly hyperpolarizing GABA(B) mediated dendritic inhibition, allowing the burst dynamics to fall under synaptic regulation. In contrast, a weaker, shunting inhibition due to GABA(A) mediated dendritic inhibition can regulate the dendritic spike waveform to decrease the dendro-somatic current flow and the resulting DAP. We therefore show that the qualitative behaviour of an individual cell can depend on the degree of synaptic input, and the exact timing of events across the spatial extent of the neuron. Thus, our results serve to illustrate the complex dynamics that can be observed in cells with significant dendritic arborisation, a nearly ubiquitous adaptation amongst principal neurons.

Population coding by electrosensory neurons

Sensory stimuli typically activate many receptors at once and therefore should lead to increases in correlated activity among central neurons. Such correlated activity could be a critical feature in the encoding and decoding of information in central circuits. Here we characterize correlated activity in response to two biologically relevant classes of sensory stimuli in the primary electrosensory nuclei, the electrosensory lateral line lobe, of the weakly electric fish Apteronotus leptorhynchus. Our results show that these neurons can display significant correlations in their baseline activities that depend on the amount of receptive field overlap. A detailed analysis of spike trains revealed that correlated activity resulted predominantly from a tendency to fire synchronous or anti-synchronous bursts of spikes. We also explored how different stimulation protocols affected correlated activity: while prey-like stimuli increased correlated activity, conspecific-like stimuli decreased correlated activity. We also computed the correlations between the variabilities of each neuron to repeated presentations of the same stimulus (noise correlations) and found lower amounts of noise correlation for communication stimuli. Therefore the decrease in correlated activity seen with communication stimuli is caused at least in part by reduced noise correlations. This differential modulation in correlated activity occurred because of changes in burst firing at the individual neuron level. Our results show that different categories of behaviorally relevant input will differentially affect correlated activity. In particular, we show that the number of correlated bursts within a given time window could be used by postsynaptic neurons to distinguish between both stimulus categories.

Control of neuronal firing by dynamic parallel fiber feedback:implications for electrosensory reafference suppression

The cancellation of self-generated components of sensory inputs is a key function of sensory feedback pathways. In many systems, cerebellar parallel fiber feedback mediates this cancellation through anti-Hebbian plasticity,resulting in the generation of a negative image of the reafferent inputs. Parallel fiber feedback involves direct excitation and disynaptic inhibition as well as synaptic plasticity on multiple time scales. How the dynamics of these processes interact with anti-Hebbian plasticity to shape synaptic inputs and provide a cancellation mechanism remains unclear. In the present study, we investigated the influence of parallel fiber feedback onto pyramidal neurons of the electrosensory lateral line lobe (ELL) in weakly electric fish under open loop conditions. We mimicked naturalistic parallel fiber inputs in an ELL brain slice by implementing an experimentally based model of this synaptic pathway using dynamic clamp. We showed that as parallel fiber activity increases, the effective input to ELL pyramidal neurons changes from net excitation to net inhibition, resulting in a non-monotonic firing response. Using a model neuron, we found that this robust non-monotonic response is due to a shift from balanced excitation and inhibition at low parallel fiber input rates, to dominant inhibition at high input rates. We then showed that this non-monotonic response provides a simple basis for negative image generation. Through changes in the mean activation rate of parallel fibers, the feedback can switch roles between enhancement and suppression of sensory inputs in a manner that is directly determined by the slope of the non-monotonic response curve.

Interval coding. I. Burst interspike intervals as indicators of stimulus intensity

Short interspike intervals such as those that occur during burst firing are hypothesized to be distinct features of the neural code. Although a number of correlations between the occurrence of burst events and aspects of the stimulus have been identified, the relationship between burst characteristics and information transfer is uncertain. Pyramidal cells in the electrosensory lobe of the weakly electric fish, Apteronotus leptorhynchus, respond to dynamic broadband electrosensory stimuli with bursts and isolated spikes. In the present study, we mimic synaptic input during sensory stimulation by direct stimulation of electrosensory pyramidal cells with broadband current in vitro. The pyramidal cells respond to this stimulus with burst interspike intervals (ISIs) that are reliably and precisely correlated with the intensity of stimulus upstrokes. We found burst ISIs must differ by a minimum of 2 ms to discriminate, with low error, differences in stimulus intensity. Based on these results, we define and quantify a candidate interval code for the processing of sensory input. Finally, we demonstrate that interval coding is restricted to short ISIs such as those generated in burst events and that the proposed interval code is distinct from rate and timing codes.

Interval coding. II. Dendrite-dependent mechanisms

The rich temporal structure of neural spike trains provides multiple dimensions to code dynamic stimuli. Popular examples are spike trains from sensory cells where bursts and isolated spikes can serve distinct coding roles. In contrast to analyses of neural coding, the cellular mechanics of burst mechanisms are typically elucidated from the neural response to static input. Bridging the mechanics of bursting with coding of dynamic stimuli is an important step in establishing theories of neural coding. Electrosensory lateral line lobe (ELL) pyramidal neurons respond to static inputs with a complex dendrite-dependent burst mechanism. Here we show that in response to dynamic broadband stimuli, these bursts lack some of the electrophysiological characteristics observed in response to static inputs. A simple leaky integrate-and-fire (LIF)-style model with a dendrite-dependent depolarizing afterpotential (DAP) is sufficient to match both the output statistics and coding performance of experimental spike trains. We use this model to investigate a simplification of interval coding where the burst interspike interval (ISI) codes for the scale of a canonical upstroke rather than a multidimensional stimulus feature. Using this stimulus reduction, we compute a quantization of the burst ISIs and the upstroke scale to show that the mutual information rate of the interval code is maximized at a moderate DAP amplitude. The combination of a reduced description of ELL pyramidal cell bursting and a simplification of the interval code increases the generality of ELL burst codes to other sensory modalities.

Muscarinic receptors control frequency tuning through the downregulation of an A-type potassium current

The functional role of cholinergic input in the modulation of sensory responses was studied using a combination of in vivo and in vitro electrophysiology supplemented by mathematical modeling. The electrosensory system of weakly electric fish recognizes different environmental stimuli by their unique alteration of a self-generated electric field. Variations in the patterns of stimuli are primarily distinguished based on their frequency. Pyramidal neurons in the electrosensory lateral line lobe (ELL) are often tuned to respond to specific input frequencies. Alterations in the tuning of the pyramidal neurons may allow weakly electric fish to preferentially select for certain stimuli. Here we show that muscarinic receptor activation in vivo enhances the excitability, burst firing, and subsequently the response of pyramidal cells to naturalistic sensory input. Through a combination of in vitro electrophysiology and mathematical modeling, we reveal that this enhanced excitability and bursting likely results from the down-regulation of an A-type potassium current. Further, we provide an explanation of the mechanism by which these currents can mediate frequency tuning.

Firing dynamics of cerebellar purkinje cells

Knowledge of intrinsic neuronal firing dynamics is a critical first step to establishing an accurate biophysical model of any neuron. In this study we examined cerebellar Purkinje cells to determine the bifurcations likely to underlie firing dynamics within a biophysically realistic and experimentally supported model. We show that Purkinje cell dynamics are consistent with a system undergoing a saddle-node bifurcation of fixed points in the transition from rest to firing and a saddle homoclinic bifurcation from firing to rest. Our analyses account for numerous observed Purkinje cell firing properties that include bistability, plateau potentials, specific aspects of the frequency-current (F-I) relationship, first spike latency, and the ability for climbing fiber input to induce state transitions in the bistable regime. We also experimentally confirm new properties predicted from our model and analysis that include the presence of a depolarizing afterpotential (DAP), the ability to fire at low frequencies (<50 Hz) and with a high gain in the F-I relationship, and a bistable region limited to low-frequency firing. Purkinje cell dynamics, including bistability, prove to arise from numerous biophysical factors that include the DAP, fast refractory dynamics, and a long membrane time constant. A hyperpolarizing activated cation current (I(H)) is shown not to be directly involved in establishing bistable dynamics but rather reduces the range for bistability. A combined electrophysiological and modeling approach thus accounts for several properties of Purkinje cells, providing a firm basis from which to assess Purkinje cell output patterns.

Neural strategies for optimal processing of sensory signals

The electrosensory system is used for both spatial navigation tasks and communication. An electric organ generates a sinusoidal electric field and cutaneous electroreceptors respond to this field. Objects such as prey or rocks cause a local low-frequency modulation of the electric field; this cue is used by electric fish for navigation and prey capture. The interference of the electric fields of conspecifics produces beats, often with high frequencies, that are also sensed by the electroreceptors; furthermore, these electric fish can transiently modulate their electric discharge as a communication signal. Thus these fish must therefore detect a variety of low-intensity signals that differ greatly in their spatial extent, frequency, and duration. Behavioral studies suggest that they are highly adapted to these tasks. Experimental and theoretical analyses of the neural circuitry for the electrosense has demonstrated many commonalities with the more common senses, e.g., topographic mapping and receptive fields with On or Off centers and surround inhibition. The integration of computational and experimental analyses has demonstrated novel mechanisms that appear to optimize weak signal detection in the electrosense including: noise shaping by correlations within single spike trains, induction of oscillations by delayed feedback inhibition, the requirement for maps with differing receptive field sizes tuned for different stimulus parameters, and the role of non-plastic feedback for adaptive cancellation of redundant signals. It is likely that these mechanisms will also be operative in other sensory systems.

A role of synchronicity of neural activity based on dynamic plasticity of synapses in encoding spatiotemporal features of electrosensory stimuli

It is quite important for investigation of sensory mechanism to understand how dynamical property of neurons is used for encoding the feature of spatiotemporally varying stimuli. To consider concretely the problem, we focus our study on electrosensory system of a weakly electric fish. Weakly electric fish generate electric field around their body using electric organ discharge (EOD) and accurately detect the location of an object through the modulation of electric field induced by the object. We made a neural network model of electrosensory lateral-line lobe (ELL). Here we show that the features of EOD modulation depending specifically distance and size of an object are encoded into the timing of burst firing of ELL neurons. These features can be represented by the spatial area of synchronous burst firing and the interburst interval in the ELL network. We show that short-term changes of excitatory and inhibitory synapses, induced by efferent signals, regulate the ELL activity so as to effectively encode the features of EOD modulation.

On the application of "equation-free modelling" to neural systems

"Equation-free modelling" is a recently-developed technique for bridging the gap between detailed, microscopic descriptions of systems and macroscopic descriptions of their collective behaviour. It uses short, repeated bursts of simulation of the microscopic dynamics to analyse the effective macroscopic equations, even though such equations are not directly available for evaluation. This paper demonstrates these techniques on a variety of networks of model neurons, and discusses the advantages and limitations of such an approach. New results include an understanding of the effects of including gap junctions in a model capable of sustaining spatially localised "bumps" of activity, and an investigation of a network of coupled bursting neurons.

Distribution and function of potassium channels in the electrosensory lateral line lobe of weakly electric apteronotid fish

Potassium channels are one of the fundamental requirements for the generation of action potentials in the nervous system, and their characteristics shape the output of neurons in response to synaptic input. We review here the distribution and function of a high-threshold potassium channel (Kv3.3) in the electrosensory lateral line lobe of the weakly electric fish Apteronotus leptorhynchus, with particular focus on the pyramidal cells in this brain structure. These cells contain both high-threshold Kv3.3 channels, as well as low-threshold potassium channels of unknown molecular identity. Kv3.3 potassium channels regulate burst discharge in pyramidal cells and enable sustained high frequency firing through their ability to reduce an accumulation of low-threshold potassium current.

Deterministic multiplicative gain control with active dendrites

Multiplicative gain control is a vital component of many theoretical analyses of neural computations, conferring the ability to scale neuronal firing rate in response to synaptic inputs. Many theories of gain control in single cells have used precisely balanced noisy inputs. Such noisy inputs can degrade signal processing. We demonstrate a deterministic method for the control of gain without the use of noise. We show that a depolarizing afterpotential (DAP), arising from active dendritic spike backpropagation, leads to a multiplicative increase in gain. Reduction of DAP amplitude by dendritic inhibition dilutes the multiplicative effect, allowing for divisive scaling of the firing rate. In contrast, somatic inhibition acts in a subtractive manner, allowing spatially distinct inhibitory inputs to perform distinct computations. The simplicity of this mechanism and the ubiquity of its elementary components suggest that many cell types have the potential to display a dendritic division of neuronal output.

Amplitude modulation of gamma band oscillations at alpha frequency produced by photic driving

Gamma band response to visual stimulation in humans has been observed to have both burst and resonance properties. Amplitude modulation of gamma activity at low frequencies has been seen in rat hippocampus and modeled in a number of forms. Significant amplitude modulation (p=0.05) of 33 Hz gamma frequency activity at the frequency of an 8 1/3 Hz photic driving stimulus, which also produced strong alpha entrainment, was observed in 67% of the channels in 42 human subjects. Similar amplitude modulation was found at a range of frequencies from greater than 50 Hz to about 28 Hz. The peak of the gamma amplitude modulation curve trailed the peak of the alpha signal by 25 to 30 ms, corresponding to a phase difference of 150 degrees to 180 degrees. The phase consistency of the gamma signal, measured across comparable times of the alpha signal, was least at the minimum amplitude modulation, and largest at the maximum. Although there was no consistent overall relation between the gamma amplitude and alpha amplitude, peak gamma amplitude values were consistently higher during post-target-stimulus alpha suppression, which occurs about 300-750 ms subsequent to stimulus presentation, than they were at the time of maximum alpha activity during the immediate post-stimulus period. It is hypothesized that there is an interaction between the alpha and gamma generating systems, in which gamma triggers alpha activity and is subsequently inhibited by it, thus producing the observed amplitude modulation. The transition from dark to light of the photic driving stimulus begins a phase resetting process in the gamma system and a concomitant burst of gamma activity; this produces an activation in the alpha system, similar to that found in the P1-N1 response in evoked potential experiments, and a subsequent inhibition of gamma production.

Dendritic Na+ current inactivation can increase cell excitability by delaying a somatic depolarizing afterpotential

Many central neurons support active dendritic spike backpropagation mediated by voltage-gated currents. Active spikes in dendrites have been shown capable of providing feedback to the soma to influence somatic excitability and firing dynamics through a depolarizing afterpotential (DAP). In pyramidal cells of the electrosensory lobe of weakly electric fish, Na(+) spikes in dendrites undergo a frequency-dependent broadening that enhances the DAP to increase somatic firing frequency. We use a combination of dynamical analysis and electrophysiological recordings to demonstrate that spike broadening in dendrites is primarily caused by a cumulative inactivation of dendritic Na(+) current. We further show that a reduction in dendritic Na(+) current increases excitability by decreasing the interspike interval and promoting burst firing. This process arises when inactivation of dendritic Na(+) current shifts the latency of the dendritic spike to delay the arrival of the DAP sufficiently to increase its impact on somatic membrane potential despite a reduction in dendritic excitability. Furthermore, the relationship between dendritic Na(+) current density and somatic excitability is nonmonotonic, as intermediate levels of dendritic Na(+) current exert the greatest excitatory influence. These results reveal that temporal shifts in dendritic spike firing provide a novel means for backpropagating spikes to influence the final output of a cell.

High-threshold K+ current increases gain by offsetting a frequency-dependent increase in low-threshold K+ current

High-frequency firing neurons are found in numerous central systems, including the auditory brainstem, thalamus, hippocampus, and neocortex. The kinetics of high-threshold K+ currents (IK(HT)) from the Kv3 subfamily has led to the proposal that these channels offset cumulative Na+ current inactivation and stabilize tonic high-frequency firing. However, all high-frequency firing neurons, examined to date, also express low-threshold K+ currents (IK(LT)) that have slower kinetics and play an important role in setting the subthreshold and filtering properties of the neuron. IK(LT) has also been shown to dampen excitability and is therefore likely to oppose high-frequency firing. In this study, we examined the role of IK(HT) in pyramidal cells of the electrosensory lobe of weakly electric fish, which are characterized by high-frequency firing, a very wide frequency range, and high levels of IK(HT). In particular, we examined the mechanisms that allow IK(HT) to set the gain of the F-I relationship by interacting with another low-threshold K+ current. We found that IK(HT) increases the gain of the F-I relationship and influences spike waveform almost exclusively in the high-frequency firing range. The frequency dependence arises from IK(HT) influencing both the IK(LT) and Na+ currents. IK(HT) thus plays a significant role in stabilizing high-frequency firing by preventing a steady-state accumulation of IK(LT) that is as important as preventing Na+ current inactivation.

Analytical solutions of the Frankenhaeuser-Huxley equations I: minimal model for backpropagation of action potentials in sparsely excitable dendrites

Hodgkin and Huxley's ionic theory of the nerve impulse embodies principles, applicable also to the impulses in vertebrate nerve fibers, as demonstrated by Bernhard Frankenhaeuser and Andrew Huxley 40 years ago. Frankenhaeuser and Huxley reformulated the classical Hodgkin-Huxley equations, in terms of electrodiffusion theory, and computed action potentials specifically for saltatory conduction in myelinated axons. In this paper, we obtain analytical solutions to the most difficult nonlinear partial differential equations in classical neurophysiology. We solve analytically the Frankenhaeuser-Huxley equations pertaining to a model of sparsely excitable, nonlinear dendrites with clusters of transiently activating, TTX-sensitive Na(+) channels, discretely distributed as point sources of inward current along a continuous (non-segmented) leaky cable structure. Each cluster or hot-spot, corresponding to a mesoscopic level description of Na(+) ion channels, includes known cumulative inactivation kinetics observed at the microscopic level. In such a third-order system, the 'recovery' variable is an electrogenic sodium-pump imbedded in the passive membrane, and the system is stabilized by the presence of a large leak conductance mediated by a composite number of ligand-gated channels permeable to monovalent cations Na(+) and K(+). In order to reproduce antidromic propagation and attenuation of action potentials, a nonlinear integral equation must be solved (in the presence of suprathreshold input, and a constant-field equation of electrodiffusion at each hot-spot with membrane gates controlling the flow of current). A perturbative expansion of the non-dimensional membrane potential (Phi) is used to obtain time-dependent analytical solutions, involving a voltage-dependent Na(+) activation (micro) and a state-dependent inactivation (eta) gating variables. It is shown that action potentials attenuate in amplitude in accordance with experimental findings, and that the spatial density distribution of transient Na(+) channels along a long dendrite contributes significantly to their discharge patterns. A major significance of the analytical modeling, in contrast to the computational modeling of backpropagating action potentials, is the provision of a continuous description of the voltage as a function of position, allowing for greater feasibility in developing large-scale biophysical neural networks, without the need for ad hoc computational modeling.

Parallel processing of sensory input by bursts and isolated spikes

Burst firing is commonly observed in many sensory systems and is proposed to transmit information reliably. Although a number of biophysical burst mechanisms have been identified, the relationship between burst dynamics and information transfer is uncertain. Electrosensory pyramidal cells have a well defined backpropagation-dependent burst mechanism. We used in vivo, in vitro, and modeling approaches to investigate pyramidal cell responses to mimics of behaviorally relevant sensory input. We found that within a given spike train, bursts are biased toward low-frequency events while isolated spikes simultaneously code for the entire frequency range. We also demonstrated that burst dynamics are essential for optimal feature detection but are not required for stimulus estimation. We conclude that burst and spike dynamics can segregate a single spike train into two parallel and complementary streams of information transfer.

A role of burst firings in encoding of spatiotemporally-varying stimulus

To investigate a role of burst firings of neurons in encoding of spatiotemporally-varying stimulus, we focus on electrosensory system of a weakly electric fish. Weakly electric fish generates electric field around its body using electric organ discharge and can accurately detect the location of an object using the modulation of electric field induced by the object. We developed a model of fish body by which we numerically describe the spatiotemporal patterns of electric field around the fish body. We also made neural models of electroreceptor distributed on the fish body and of electrosensory lateral-line lobe (ELL) to investigate what kinds of information of electric field distorted by an object they detect. Here we show that the spatiotemporal features of electric field around the fish body are encoded by the timing of burst firings of ELL neurons. The information of object distance is extracted by the area of synchronous firings of neurons in a higher nucleus, torus semicircularis.

Oscillatory burst discharge generated through conditional backpropagation of dendritic spikes

Gamma frequencies of burst discharge (>40 Hz) have become recognized in select cortical and non-cortical regions as being important in feature extraction, neural synchrony and oscillatory discharge. Pyramidal cells of the electrosensory lateral line lobe (ELL) of Apteronotus leptorhynchus generate burst discharge in relation to specific features of sensory input in vivo that resemble those recognized as gamma frequency discharge when examined in vitro. We have shown that these bursts are generated by an entirely novel mechanism termed conditional backpropagation that involves an intermittent failure of dendritic Na+ spike conduction. Conditional backpropagation arises from a frequency-dependent broadening of dendritic spikes during repetitive discharge, and a mismatch between the refractory periods of somatic and dendritic spikes. A high threshold class of K+ channel, AptKv3.3, is expressed at high levels and distributed over the entire soma-dendritic axis of pyramidal cells. AptKv3.3 channels are shown to contribute to the repolarization of both somatic and dendritic spikes, with pharmacological blockade of dendritic Kv3 channels revealing an important role in controlling the threshold for burst discharge. The entire process of conditional back-propagation and burst output is successfully simulated using a new compartmental model of pyramidal cells that incorporates a cumulative inactivation of dendritic K+ channels during repetitive discharge. This work is important in demonstrating how the success of spike backpropagation can control the output of a principle sensory neuron, and how this process is regulated by the distribution and properties of voltage-dependent K+ channels.

Insight into the mechanisms of neuronal processing from electric fish

Weakly electric fish use their electric fields to locate objects and communicate with each other. Their electric discharges vary with species, gender, and social status. This variation is mediated by steroid and peptide hormones that influence ion currents through changes in gene expression or phosphorylation state. Understanding how electric fish decode the perturbations of their electric fields that result from interactions with the discharges of other fish or prey is illuminating general mechanisms of neuronal processing. Their central sensory circuits are specialized to process amplitude modulated signals, to detect microsecond variations in spike timing, and are dynamically reconfigured depending on the stimulus parameters.

Inactivation of Kv3.3 potassium channels in heterologous expression systems

Kv3.3 K+ channels are believed to incorporate an NH2-terminal domain to produce an intermediate rate of inactivation relative to the fast inactivating K+ channels Kv3.4 and Kv1.4. The rate of Kv3.3 inactivation has, however, been difficult to establish given problems in obtaining consistent rates of inactivation in expression systems. This study characterized the properties of AptKv3.3, the teleost homologue of Kv3.3, when expressed in Chinese hamster ovary (CHO) or human embryonic kidney (HEK) cells. We show that the properties of AptKv3.3 differ significantly between CHO and HEK cells, with the largest difference occurring in the rate and voltage dependence of inactivation. While AptKv3.3 in CHO cells showed a fast and voltage-dependent rate of inactivation consistent with N-type inactivation, currents in HEK cells showed rates of inactivation that were voltage-independent and more consistent with a slower C-type inactivation. Examination of the mRNA sequence revealed that the first methionine start site had a weak Kozak consensus sequence, suggesting that the lack of inactivation in HEK cells could be due to translation at a second methionine start site downstream of the NH2-terminal coding region. Mutating the nucleotide sequence surrounding the first methionine start site to one more closely resembling a Kozak consensus sequence produced currents that inactivated with a fast and voltage-dependent rate of inactivation in both CHO and HEK cells. These results indicate that under the appropriate conditions Kv3.3 channels can exhibit fast and reliable inactivation that approaches that more typically expected of "A"-type K+ currents.

Periodic forcing of a model sensory neuron

We study the effects of sinusoidally modulating the current injected into a model sensory neuron from the weakly electric fish Apteronotus leptorhynchus. This neuron's behavior is known to switch from quiescence to periodic firing to bursting as the injected current is increased. The bifurcation separating periodic from bursting behavior is a saddle-node bifurcation of periodic orbits, and it has been shown previously that there is "type-I burst excitability" associated with this bifurcation, similar to the usual excitability associated with the transition from quiescence to periodic firing. Here we show numerically that sinusoidal modulation of the dc current injected into the model neuron can switch it from periodic to burst firing, or vice versa, depending on the frequency of modulation and the distance to the burst excitability threshold. This is explained by mapping resonance tongues in parameter space. We also show that such a model neuron can undergo stochastic resonance near the transition from periodic to burst firing, as a result of the burst excitability, regardless of the location (soma or dendrite) of the signal and noise. The novelty is that the "output event" is now a burst rather than a single action potential, and the neuron returns to almost periodic firing between bursts, rather than to the vicinity of a fixed point. Since the neuron under study is a sensory neuron that must encode signals with varying temporal structure in the presence of considerable intrinsic noise, these aspects are of potential importance to electrosensory processing and also to other bursting neurons that have periodic input.

Gamma rhythmic bursts: coherence control in networks of cortical pyramidal neurons

Much evidence indicates that synchronized gamma-frequency (20-70 Hz) oscillation plays a significant functional role in the neocortex and hippocampus. Chattering neuron is a possible neocortical pacemaker for the gamma oscillation. Based on our recent model of chattering neurons, here we study how gamma-frequency bursting is synchronized in a network of these neurons. Using a phase oscillator description, we first examine how two coupled chattering neurons are synchronized. The analysis reveals that an incremental change of the bursting mode, such as from singlet to doublet, always accompanies a rapid transition from antisynchronous to synchronous firing. The state transition occurs regardless of what changes the bursting mode. Within each bursting mode, the neuronal activity undergoes a gradual change from synchrony to antisynchrony. Since the sensitivity to Ca(2+) and the maximum conductance of Ca(2+)-dependent cationic current as well as the intensity of input current systematically control the bursting mode, these quantities may be crucial for the regulation of the coherence of local cortical activity. Numerical simulations demonstrate that the modulations of the calcium sensitivity and the amplitude of the cationic current can induce rapid transitions between synchrony and asynchrony in a large-scale network of chattering neurons. The rapid synchronization of chattering neurons is shown to synchronize the activities of regular spiking pyramidal neurons at the gamma frequencies, as may be necessary for selective attention or binding processing in object recognition.

Type I burst excitability

We introduce the concept of "type I burst excitability", which is a generalization of the "normal" excitability that is well-known in cardiac and neural systems. We demonstrate this type of burst excitability in a specific model system, a pyramidal cell from the electrosensory lateral line lobe of the weakly electric fish Apteronotus leptorhynchus. As depolarizing current is increased, a saddle-node bifurcation of periodic orbits occurs, which separates tonic and burst activity. This bifurcation is responsible for the excitable nature of the system, and is the basis for the "type I" designation. We verify the existence of this transition from in vitro recordings of a number of actual pyramidal cells. A scaling relationship between the magnitude and duration of a current pulse required to induce a burst is derived. We also observe this type of burst excitability and the scaling relationships in a multicompartmental model that is driven by realistic stochastic synaptic inputs mimicking sensory input. We conclude by discussing the relevance of burst excitability to communication between weakly electric fish.

A dynamic dendritic refractory period regulates burst discharge in the electrosensory lobe of weakly electric fish

Na+-dependent spikes initiate in the soma or axon hillock region and actively backpropagate into the dendritic arbor of many central neurons. Inward currents underlying spike discharge are offset by outward K+ currents that repolarize a spike and establish a refractory period to temporarily prevent spike discharge. We show in a sensory neuron that somatic and dendritic K+ channels differentially control burst discharge by regulating the extent to which backpropagating dendritic spikes can re-excite the soma. During repetitive discharge a progressive broadening of dendritic spikes promotes a dynamic increase in dendritic spike refractory period. A leaky integrate-and-fire model shows that spike bursts are terminated when a decreasing somatic interspike interval and an increasing dendritic spike refractory period synergistically act to block backpropagation. The time required for the somatic interspike interval to intersect with dendritic refractory period determines burst frequency, a time that is regulated by somatic and dendritic spike repolarization. Thus, K+ channels involved in spike repolarization can efficiently control the pattern of spike output by establishing a soma-dendritic interaction that invokes dynamic shifts in dendritic spike properties.

Persistent Na+ current modifies burst discharge by regulating conditional backpropagation of dendritic spikes

The estimation and detection of stimuli by sensory neurons is affected by factors that govern a transition from tonic to burst mode and the frequency characteristics of burst output. Pyramidal cells in the electrosensory lobe of weakly electric fish generate spike bursts for the purpose of stimulus detection. Spike bursts are generated during repetitive discharge when a frequency-dependent broadening of dendritic spikes increases current flow from dendrite to soma to potentiate a somatic depolarizing afterpotential (DAP). The DAP eventually triggers a somatic spike doublet with an interspike interval that falls inside the dendritic refractory period, blocking spike backpropagiation and the DAP. Repetition of this process gives rise to a rhythmic dendritic spike failure, termed conditional backpropagation, that converts cell output from tonic to burst discharge. Through in vitro recordings and compartmental modeling we show that burst frequency is regulated by the rate of DAP potentiation during a burst, which determines the time required to discharge the spike doublet that blocks backpropagation. DAP potentiation is magnified through a positive feedback process when an increase in dendritic spike duration activates persistent sodium current (I(NaP)). I(NaP) further promotes a slow depolarization that induces a shift from tonic to burst discharge over time. The results are consistent with a dynamical systems analysis that shows that the threshold separating tonic and burst discharge can be represented as a saddle-node bifurcation. The interaction between dendritic K(+) current and I(NaP) provides a physiological explanation for a variable time scale of bursting dynamics characteristic of such a bifurcation.

Bursting neurons signal input slope

Brief bursts of high-frequency action potentials represent a common firing mode of pyramidal neurons, and there are indications that they represent a special neural code. It is therefore of interest to determine whether there are particular spatial and temporal features of neuronal inputs that trigger bursts. Recent work on pyramidal cells indicates that bursts can be initiated by a specific spatial arrangement of inputs in which there is coincident proximal and distal dendritic excitation (Larkum et al., 1999). Here we have used a computational model of an important class of bursting neurons to investigate whether there are special temporal features of inputs that trigger bursts. We find that when a model pyramidal neuron receives sinusoidally or randomly varying inputs, bursts occur preferentially on the positive slope of the input signal. We further find that the number of spikes per burst can signal the magnitude of the slope in a graded manner. We show how these computations can be understood in terms of the biophysical mechanism of burst generation. There are several examples in the literature suggesting that bursts indeed occur preferentially on positive slopes (Guido et al., 1992; Gabbiani et al., 1996). Our results suggest that this selectivity could be a simple consequence of the biophysics of burst generation. Our observations also raise the possibility that neurons use a burst duration code useful for rapid information transmission. This possibility could be further examined experimentally by looking for correlations between burst duration and stimulus variables.

Noise-induced divisive gain control in neuron models

A recent computational study of gain control via shunting inhibition has shown that the slope of the frequency-versus-input (f-I) characteristic of a neuron can be decreased by increasing the noise associated with the inhibitory input (Neural Comput. 13, 227-248). This novel noise-induced divisive gain control relies on the concommittant increase of the noise variance with the mean of the total inhibitory conductance. Here we investigate this effect using different neuronal models. The effect is shown to occur in the standard leaky integrate-and-fire (LIF) model with additive Gaussian white noise, and in the LIF with multiplicative noise acting on the inhibitory conductance. The noisy scaling of input currents is also shown to occur in the one-dimensional theta-neuron model, which has firing dynamics, as well as a large scale compartmental model of a pyramidal cell in the electrosensory lateral line lobe of a weakly electric fish. In this latter case, both the inhibition and the excitatory input have Poisson statistics; noise-induced divisive inhibition is thus seen in f-I curves for which the noise increases along with the input I. We discuss how the variation of the noise intensity along with inputs is constrained by the physiological context and the class of model used, and further provide a comparison of the divisive effect across models.

Ghostbursting: a novel neuronal burst mechanism

Pyramidal cells in the electrosensory lateral line lobe (ELL) of weakly electric fish have been observed to produce high-frequency burst discharge with constant depolarizing current (Turner et al., 1994). We present a two-compartment model of an ELL pyramidal cell that produces burst discharges similar to those seen in experiments. The burst mechanism involves a slowly changing interaction between the somatic and dendritic action potentials. Burst termination occurs when the trajectory of the system is reinjected in phase space near the "ghost" of a saddle-node bifurcation of fixed points. The burst trajectory reinjection is studied using quasi-static bifurcation theory, that shows a period doubling transition in the fast subsystem as the cause of burst termination. As the applied depolarization is increased, the model exhibits first resting, then tonic firing, and finally chaotic bursting behavior, in contrast with many other burst models. The transition between tonic firing and burst firing is due to a saddle-node bifurcation of limit cycles. Analysis of this bifurcation shows that the route to chaos in these neurons is type I intermittency, and we present experimental analysis of ELL pyramidal cell burst trains that support this model prediction. By varying parameters in a way that changes the positions of both saddle-node bifurcations in parameter space, we produce a wide gallery of burst patterns, which span a significant range of burst time scales.

Function of NMDA receptors and persistent sodium channels in a feedback pathway of the electrosensory system

Voltage-dependent amplification of ionotropic glutamatergic excitatory postsynaptic potentials (EPSPs) can, in many vertebrate neurons, be due either to the intrinsic voltage dependence of N-methyl-D-aspartate (NMDA) receptors, or voltage-dependent persistent sodium channels expressed on postsynaptic dendrites or somata. In the electrosensory lateral line lobe (ELL) of the gymnotiform fish Apteronotus leptorhynchus, glutamatergic inputs onto pyramidal cell apical dendrites provide a system where both amplification mechanisms are possible. We have now examined the roles for both NMDA receptors and sodium channels in the control of EPSP amplitude at these synapses. An antibody specific for the A. leptorhynchus NR1 subunit reacted strongly with ELL pyramidal cells and were particularly abundant in the spines of pyramidal cell apical dendrites. We have also shown that NMDA receptors contributed strongly to the late phase of EPSPs evoked by stimulation of the feedback fibers terminating on the apical dendritic spines; further, these EPSPs were voltage dependent. Blockade of NMDA receptors did not, however, eliminate the voltage dependence of these EPSPs. Blockade of somatic sodium channels by local somatic ejection of tetrodotoxin (TTX), or inclusion of QX314 (an intracellular sodium channel blocker) in the recording pipette, reduced the evoked EPSPs and completely eliminated their voltage dependence. We therefore conclude that, in the subthreshold range, persistent sodium currents are the main contributor to voltage-dependent boosting of EPSPs, even when they have a large NMDA receptor component.