Analytical reconstruction of the neuronal input current from spike train data

The time course of the current driving action potential generation at a neuron investigated experimentally is in general not measurable directly. In this paper an indirect method is introduced that allows estimation of this unknown current time course using only spike train data. Assuming the leaky integrator model as valid for the action potential encoding site of the investigated neuron, the unknown input current is obtained by determining (analytically) a current time course that upon injection into the leaky integrator model evokes action potential sequences identical to those observed experimentally. Applications of this current-reconstruction procedure to neuronal output data obtained from a leaky integrator model showed that the procedure allows a good estimation of the underlying input current even if the membrane time constant of the investigated neuron is not known exactly. Additionally, an application of current reconstruction to experimental data obtained from a cat muscle spindle primary afferent subject to repeated -stimuli is demonstrated.     

The adaptation ability of neuronal models subject to a current step stimulus

Three neuronal models of the spike initiating process were investigated with respect to their ability to show adaptation to a current step: (i) the perfect integrator model (PIM), (ii) the leaky integrator model (LIM), and (iii) the Hodgkin-Huxley (HH-) model. It was found that although each neuronal model will generate different response spike trains to a given stimulus, all responses fulfilled the criteria of a deterministic neural response (Awiszus 1988). The results show that both PIM and LIM are unable to show adaptation regardless of the choice of model parameters whereas the HH-model shows a clear rate of discharge adaptation. The reason for this adaptation lies in the fact that there are conditions for the HH-model where a step stimulus is highly effective. These conditions are investigated by means of a phase plane analysis. Consequences of these results for the explanation of neuronal adaptation and the validity of the neuronal models investigated are discussed.     

Nonuniformity in the linear network model of the oculomotor integrator produces approximately fractional-order dynamics and more realistic neuron behavior

The oculomotor integrator is a network that is composed of neurons in the medial vestibular nuclei and nuclei prepositus hypoglossi in the brainstem. Those neurons act approximately as fractional integrators of various orders, converting eye velocity commands into signals that are intermediate between velocity and position. The oculomotor integrator has been modeled as a network of linear neural elements, the time constants of which are lengthened by positive feedback through reciprocal inhibition. In this model, in which each neuron reciprocally inhibits its neighbors with the same Gaussian profile, all model neurons behave as identical, first-order, low-pass filters with dynamics that do not match the variable, approximately fractional-order dynamics of the neurons that compose the actual oculomotor integrator. Fractional-order integrators can be approximated by weighted sums of first-order, low-pass filters with diverse, broadly distributed time constants. Dynamic systems analysis reveals that the model integrator indeed has many broadly distributed time constants. However, only one time constant is expressed in the model due to the uniformity of its network connections. If the model network is made nonuniform by removing the reciprocal connections to and from a small number of neurons, then many more time constants are expressed. The dynamics of the neurons in the nonuniform network model are variable, approximately fractional-order, and resemble those of the neurons that compose the actual oculomotor integrator. Completely removing the connections to and from a neuron is equivalent to eliminating it, an operation done previously to demonstrate the robustness of the integrator network model. Ironically, the resulting nonuniform network model, previously supposed to represent a pathological integrator, may in fact represent a healthy integrator containing neurons with realistically variable, approximately fractional-order dynamics. Received: 8 August 1997 / Accepted in revised form: 18 June 1998     

Exploring optimal current stimuli that provide membrane voltage tracking in a neuron model

Studying neurons from an energy efficiency perspective has produced results in the research literature. This paper presents a method that enables computation of low energy input current stimuli that are able to drive a reduced Hodgkin–Huxley neuron model to approximate a prescribed time-varying reference membrane voltage. An optimal control technique is used to discover an input current that optimally minimizes a user selected balance between the square of the input stimulus current (input current ‘energy’) and the difference between the reference voltage and the membrane voltage (tracking error) over a stimulation period. Selecting reference signals to be membrane voltages produced by the neuron model in response to common types of input currents i(t) enables a comparison between i(t) and the determined optimal current stimulus i*(t). The intent is not to modify neuron dynamics, but through comparison of i(t) and i*(t) provide insight into neuron dynamics. Simulation results for four different bifurcation types demonstrate that this method consistently finds lower energy stimulus currents i*(t) that are able to approximate membrane voltages as produced by higher energy input currents i(t) in this neuron model.     

Safety evaluation <Emphasis Type="Italic">in vitro</Emphasis> of <Emphasis Type="Italic">Enterococcus durans</Emphasis> from Tibetan traditional fermented yak milk

Despite its ubiquity in fermented dairy products, the safety of lactic acid enterococcal bacteria remains controversial. In this study, five Enterococcus durans strains — A1, A2, B1, B2, and C1 — were isolated from traditional fermented yak milk from Tibet. To evaluate the strains’ safety, biogenic amine production, antibiotic resistance and presence of known virulence determinants were investigated. Strain A1 can produce biogenic amines for histamine, spermine, and spermidine (mean values: 8.64, 8.31, and 0.30 mg/L, respectively). Polymerase chain reaction amplification for Strain A1 found genes involved in expression of gelatinase (gleE), cytolysin (cylA, cylB, and cylM), sex pheromones (ccf and cpd) and cell wall adhesion (efaA). Strain A2 showed sensitivity or intermediate resistance to all tested antibiotics, and no virulence determinants except gelE and ccf, but did produce tyramine at a relatively high level (912.02 mg/L). Both strains B1 and B2 could produce histamine (10.43 and 10.56 mg/L, respectively), and showed vancomycin resistance; B1 also produced tyramine (504.02 mg/L). Strain C1 could produce all five biogenic amines tested in the study -putrescine, histamine, tyramine, spermine, and spermidine; concentrations were 6.51, 9.59, 205.85, 5.55, and 5.39 mg/L, respectively. All E. durans strains found in Tibetan traditional fermented yak milk thus offer potential risk.     

Accuracy and response-time distributions for decision-making: linear perfect integrators versus nonlinear attractor-based neural circuits

Animals choose actions based on imperfect, ambiguous data. “Noise” inherent in neural processing adds further variability to this already-noisy input signal. Mathematical analysis has suggested that the optimal apparatus (in terms of the speed/accuracy trade-off) for reaching decisions about such noisy inputs is perfect accumulation of the inputs by a temporal integrator. Thus, most highly cited models of neural circuitry underlying decision-making have been instantiations of a perfect integrator. Here, in accordance with a growing mathematical and empirical literature, we describe circumstances in which perfect integration is rendered suboptimal. In particular we highlight the impact of three biological constraints: (1) significant noise arising within the decision-making circuitry itself; (2) bounding of integration by maximal neural firing rates; and (3) time limitations on making a decision. Under conditions (1) and (2), an attractor system with stable attractor states can easily best an integrator when accuracy is more important than speed. Moreover, under conditions in which such stable attractor networks do not best the perfect integrator, a system with unstable initial states can do so if readout of the system’s final state is imperfect. Ubiquitously, an attractor system with a nonselective time-dependent input current is both more accurate and more robust to imprecise tuning of parameters than an integrator with such input. Given that neural responses that switch stochastically between discrete states can “masquerade” as integration in single-neuron and trial-averaged data, our results suggest that such networks should be considered as plausible alternatives to the integrator model.     

Spatial and temporal correlations of spike trains in frog retinal ganglion cells

For a neuron, firing activity can be in synchrony with that of others, which results in spatial correlation; on the other hand, spike events within each individual spike train may also correlate with each other, which results in temporal correlation. In order to investigate the relationship between these two phenomena, population neurons’ activities of frog retinal ganglion cells in response to binary pseudo-random checker-board flickering were recorded via a multi-electrode recording system. The spatial correlation index (SCI) and temporal correlation index (TCI) were calculated for the investigated neurons. Statistical results showed that, for a single neuron, the SCI and TCI values were highly related—a neuron with a high SCI value generally had a high TCI value, and these two indices were both associated with burst activities in spike train of the investigated neuron. These results may suggest that spatial and temporal correlations of single neuron’s spiking activities could be mutually modulated; and that burst activities could play a role in the modulation. We also applied models to test the contribution of spatial and temporal correlations for visual information processing. We show that a model considering spatial and temporal correlations could predict spikes more accurately than a model does not include any correlation.     

On the integration of subthreshold inputs from Perforant Path and Schaffer Collaterals in hippocampal CA1 pyramidal neurons

Using a realistic model of a CA1 hippocampal pyramidal neuron, we make experimentally testable predictions on the roles of the non-specific cation current, I _h , and the A-type Potassium current, I _A , in modulating the temporal window for the integration of the two main excitatory afferent pathways of a CA1 neuron, the Schaffer Collaterals and the Perforant Path. The model shows that the experimentally observed increase in the dendritic density of I _h and I _A could have a major role in constraining the temporal integration window for these inputs, in such a way that a somatic action potential (AP) is elicited only when they are activated with a relative latency consistent with the anatomical arrangement of the hippocampal circuitry.     

Continuous functions for the analysis of sensory transduction

Sensory transduction at a primary receptor neuron yields a current that drives the generation of action potentials. Due to the inaccessibility of that current for direct measurements the analysis of sensory transduction requires the use of neuronal output functions that give an indirect measure for the input current, i.e. the current at the impulse initiating site. Three continuous neuronal output functions are investigated with respect to their ability to reconstruct the input current (i) the membrane potential recorded under sodium channel block referred to as receptor potential, (ii) the interspike-interval function (Awiszus 1988a) and (iii) the phase lag function which is introduced in this paper. The behaviour of these three functions for constant and dynamically varying input is studied at the Hodgkin-Huxley model (Hodgkin and Huxley 1952) because for this model neuron it is possible to compare the input current estimates obtained from the output functions with the true input current. It was found that for constant and for sufficiently slow varying input all three functions allow a valid reconstruction of the input current time course. On the other hand, if the input current changes rapidly all three estimated input current time courses show considerable deviations from the true time course. The largest maximal deviation is shown by the current estimate obtained from the receptor potential whereas the phase lag function yields the smallest input current misjudgement. An experimental example to illustrate the procedure to obtain the phase lag function for a muscle spindle primary afferent is given.     

Analysis of the effects of modulatory agents on a modeled bursting neuron: Dynamic interactions between voltage and calcium dependent systems

In a computational model of the bursting neuron R15, we have implemented proposed mechanisms for the modulation of two ionic currents (I _R andI _SI) that play key roles in regulating its spontaneous electrical activity. The model was sufficient to simulate a wide range of endogenous activity in the presence of various concentrations of serotonin (5-HT) or dopamine (DA). The model was also sufficient to simulate the responses of the neuron to extrinsic current pulses and the ways in which those responses were altered by 5-HT or DA. The results suggest that the actions of modulatory agents and second messengers on this neuron, and presumably other neurons, cannot be understood on the basis of their direct effects alone. It is also necessary to take into account the indirect effects of these agents on other unmodulated ion channels. These indirect effects occur through the dynamic interactions of voltage-dependent and calcium-dependent processes.     

Impact of slow K<Superscript>+</Superscript> currents on spike generation can be described by an adaptive threshold model

A neuron that is stimulated by rectangular current injections initially responds with a high firing rate, followed by a decrease in the firing rate. This phenomenon is called spike-frequency adaptation and is usually mediated by slow K⁺ currents, such as the M-type K⁺ current (I _M ) or the Ca²⁺-activated K⁺ current (I _AHP ). It is not clear how the detailed biophysical mechanisms regulate spike generation in a cortical neuron. In this study, we investigated the impact of slow K⁺ currents on spike generation mechanism by reducing a detailed conductance-based neuron model. We showed that the detailed model can be reduced to a multi-timescale adaptive threshold model, and derived the formulae that describe the relationship between slow K⁺ current parameters and reduced model parameters. Our analysis of the reduced model suggests that slow K⁺ currents have a differential effect on the noise tolerance in neural coding.     

An Ionic Current Model for Medullary Respiratory Neurons

Neurons of the mammalian medullary respiratory center have complex patterns of electrophysiological behavior. Three typical phenomena associated with these patterns are spike frequency adaptation (SFA), delayed excitation (DE), and postinhibitory rebound (PIR). Although several nuclei are associated with the medullary-pontine respiratory center, we focused on neurons from two nuclei: (1) the ventral subnucleus of the nucleus tractus solitarius (vNTS) of the dorsal respiratory group and (2) the nucleus ambiguus (NA) of the ventral respiratory group. We developed a Hodgkin-Huxley (HH) type model of the typical medullary neuron that is capable of mimicking the discharge pattern of real neurons to a very high degree. Closer examination of typical data revealed, however, that there was not one type of medullary respiratory neuron, but at least three (types A, B ₁, and B ₂). We classified these neurons based on the electrophysiologic phenomena that they exhibited (type A exhibits DE but not PIR; types B ₁ and B ₂ exhibit PIR but not DE; all types are adapting). Our objective was to relate each of these well-known phenomena to specific ionic current mechanisms. In the model, three currents directly affect the phenomena investigated: the Ca²⁺-activated K ⁺ current, I _K,Ca , controls peak and steady-state firing rates and the time constant of adaptation; the transient outward K ⁺ current, I _A, is responsible for all aspects of DE, including the dependence of delay on the magnitude and duration of conditioning hyperpolarization; and the hyperpolarization-activated current, I _h, elicits PIR and dictates its dependencies. We consider that our HH model represents a unifying structure, whereby different electrophysiological phenomena or discharge patterns can be emulated using different strengths of the component ionic membrane currents (particularly I _K,Ca , I _A, and I _h). Moreover, its predictions suggest that the electrophysiological characteristics of medullary respiratory neurons, from different areas of the brainstem and even from different species, can be modeled using the same structural framework, wherein the specific properties of individual neurons are emulated by adjusting the strengths of key ionic membrane currents in the model.     

The effect of cooling on the acetylcholine-induced current of identified <Emphasis Type="Italic">Helix pomatia</Emphasis> Br neuron

The Br neuron of the snail Helix pomatia, involved in neuronal regulation of various homeostatic and adaptive mechanisms, represents an interesting model for studying effects of temperature changes on neuronal activity of poikilotherms. The acetylcholine (ACh) induces a transient, inward dose-dependent current in the identified Br neuron. In the work presented, we analyses the effects of cooling on the ACh-induced inward current. The amplitude of ACh-induced inward current was markedly decreased after cooling and the speed of the decay of ACh response was decreased. Sensitivity to cooling of Ach-activated current on the Br neuron is mediated by a mechanism that does not involve change in the apparent receptor affinity or the cooperativity of binding.     

Use of Long-Term Stored Vector Information in the Neotropical Ant <Emphasis Type="BoldItalic">Gigantiops destructor</Emphasis>

We investigated how the formicine ant Gigantiops destructor can use vector information to navigate within the cluttered environment of the rain forest. Displaced foragers use skylight information to move in the theoretical feeder-to-nest direction, whether they are prevented from updating their path-integrator during foraging or captured at the departure from their nest, i.e. with a current accumulator state very close to zero. Only ants that have collected food are able to download a long-term stored reference vector pointing in the nest direction, irrespective of the current accumulator state of their path-integrator stored in a working memory and independent of familiar landmarks. Depending on the release sites, ants that became lost at a maximum distance of 50 cm could still hit and recognize their familiar route, or they engaged in a systematic search for it centered on the release sites. In contrast to Cataglyphis desert ants, Gigantiops ants do not rely primarily on the current accumulator state of their egocentric path integrator. Such a long-term vector-based navigation primed by food capture is well adapted for a tropical ant foraging during periods spanning several hours. This could prevent the numerous cumulative errors in the evaluation of the angles steered that might result from a continuously running path-integrator operating during complex foraging patterns performed at ground or arboreal levels and during passive displacement in response to heavy rain.     

Study on dynamic characteristics’ change of hippocampal neuron reduced models caused by the Alzheimer’s disease

In the paper, based on the electrophysiological experimental data, the Hippocampal neuron reduced model under the pathology condition of Alzheimer’s disease (AD) has been built by modifying parameters’ values. The reduced neuron model’s dynamic characteristics under effect of AD are comparatively studied. Under direct current stimulation, compared with the normal neuron model, the AD neuron model’s dynamic characteristics have obviously been changed. The neuron model under the AD condition undergoes supercritical Andronov–Hopf bifurcation from the rest state to the continuous discharge state. It is different from the neuron model under the normal condition, which undergoes saddle-node bifurcation. So, the neuron model changes into a resonator with monostable state from an integrator with bistable state under AD’s action. The research reveals the neuron model’s dynamic characteristics’ changing under effect of AD, and provides some theoretic basis for AD research by neurodynamics theory.     

Dynamics of one-dimensional spiking neuron models

In this paper we make a rigorous mathematical analysis of one-dimensional spiking neuron models in a unified framework. We find that, under conditions satisfied in particular by the periodically and aperiodically driven leaky integrator as well as some of its variants, the spike map is increasing on its range, which leaves no room for chaotic behavior. A rigorous expression of the Lyapunov exponent is derived. Finally, we analyse the periodically driven perfect integrator and show that the restriction of the phase map to its range is always conjugated to a rotation, and we provide an explicit expression of the invariant measure.     

Mechanisms of Generation of Pacemaker Activity by Identified Helix Neurons

We studied the mechanisms of generation of pacemaker activity in identified neurons of Helix pomatia. For this purpose, we isolated the PPa2 and PPa7 neurons generating spontaneous rhythmic monomodal activity and PPa1 neuron with bursting activity. It was demonstrated that isolated PPa2 and PPa7 cells produce endogenous rhythmic activity that was not considerably modified by external application of 1 mM CdCl₂. Sometimes, only low-amplitude dendritic action potentials (AP) were observed instead of generation of full-amplitude somatic AP. In contrast, isolation of the PPa1 neuron eliminated its bursting activity, but subsequent application of oxytocin on this neuron recovered such activity. This finding shows that the bursting activity of the PPa1 neuron is of an exogenous nature. Application of 1 mM CdCl₂ suppressed this bursting activity, but when Cd²⁺ was applied against the background of superfusion of the neuron with Ringer solution containing a bursting activity-initiating neuropeptide obtained from the molluscan CNS, this blocker was incapable of suppressing the bursting activity. A blocker of the hyperpolarization-activated current (I _h , H current), Cs⁺ (10 mM) exerted no noticeable effect on the activity of the studied neurons. Our findings allow us to conclude that the pacemaker activity is initiated within the dendritic tree of a cell and is then electrotonically spread to the soma, where full-amplitude AP are generated. It seems probable that Ca²⁺ ions and H current are not directly involved in generation of the pacemaker activity in the studied snail neurons.     

The response of excitable membrane models to a cyclic input

The response of a space-clamped patch of Hodgkin-Huxley membrane to an applied current density ofA cos(2ft)+BA/cm² is computed for frequencies from 5 to 250 Hz. The train of action potentials generated is phase-locked to the driving cycle,N action potentials occurring at fixed phases inM cycles. For frequencies whereN/M is a simple ratio a describing function for the membrane is computed. The phase-locked behaviour and describing functions are similar to those obtained for a simple leaky integrator neurone model.     

Modulation of swimming behavior in the medicinal leech

Expression of swimming in the medicinal leech (Hirudo medicinalis) is modulated by serotonin, a naturally occurring neurohormone. Exogenous application of serotonin engenders spontaneous swimming activity in nerve-cord preparations. We examined whether this activity is due to enhanced participation of swim motor neurons (MNs) in generating the swimming rhythm. We found that depolarizing current injections into MNs during fictive swimming are more effective in shifting cycle phase in nerve cords following serotonin exposure. In such preparations, the dynamics of membrane potential excursions following current injection into neuronal somata are substantially altered. We observed: 1) a delayed outward rectification (relaxation) during depolarizing current injection, most marked in inhibitory MNs; and 2) in excitor MNs, an enhancement of postinhibitory rebound (PIR) and afterhyperpolarizing potentials (AHPs) following hyperpolarizing and depolarizing current pulses, respectively. In contrast, we found little alteration in MN properties in leech nerve cords depleted of amines. We propose that enhanced expression of swimming activity in leeches exposed to elevated serotonin is due, partly, to enhancement of relaxation, PIR and AHP in MNs. We believe that as a consequence of alterations in cellular properties and synaptic interactions (subsequent paper) by serotonin, MNs are reconfigured to more effectively participate in generating and expressing the leech swimming rhythm.Abbreviations AHP Afterhyperpolarizing potential - DCC Discontinuous current clamp - DE Dorsal excitor motor neuron - DI Dorsal inhibitor motor neuron - IPSP Inhibitory postsynaptic potential - MN Motor neuron - PIR Postinhibitory rebound - VE Ventral excitor motor neuron - VI Ventral inhibitor motor neuron     

Spikes annihilation in the Hodgkin-Huxley neuron

The Hodgkin–Huxley (HH) neuron is a nonlinear system with two stable states: A fixed point and a limit cycle. Both of them co-exist. The behavior of this neuron can be switched between these two equilibria, namely spiking and resting respectively, by using a perturbation method. The change from spiking to resting is named Spike Annihilation, and the transition from resting to spiking is named Spike Generation. Our intention is to determine if the HH neuron in 2D is controllable (i.e., if it can be driven from a quiescent state to a spiking state and vice versa). It turns out that the general system is unsolvable.¹ In this paper, first of all,² we analytically prove the existence of a brief current pulse, which, when delivered to the HH neuron during its repetitively firing state, annihilates its spikes. We also formally derive the characteristics of this brief current pulse. We then proceed to explore experimentally, by using numerical simulations, the properties of this pulse, namely the range of time when it can be inserted (the minimum phase and the maximum phase), its magnitude, and its duration. In addition, we study the solution of annihilating the spikes by using two successive stimuli, when the first is, of its own, unable to annihilate the neuron. Finally, we investigate the inverse problem of annihilation, namely the spike generation problem, when the neuron switches from resting to firing. ¹ This conclusion is a consequence of three well-known fundamental results, namely Hilbert 16th Problem, the Poincare–Bendixon Theorem and the Hopf Bifurcation Theorem. ² We are extremely grateful to the feedback we received from the anonymous Referees to the initial version of the paper. Their comments significantly improved the quality of the current version. Thanks a lot!