Conductance-based Neuron Model Research Articles

Reliability of signal transmission in stochastic nerve axon equations

We introduce a method for computing probabilities for spontaneous activity and propagation failure of the action potential in spatially extended, conductance-based neuronal models subject to noise, based on statistical properties of the membrane potential. We compare different estimators with respect to the quality of detection, computational costs and robustness and propose the integral of the membrane potential along the axon as an appropriate estimator to detect both spontaneous activity and propagation failure. Performing a model reduction we achieve a simplified analytical expression based on the linearization at the resting potential (resp. the traveling action potential). This allows to approximate the probabilities for spontaneous activity and propagation failure in terms of (classical) hitting probabilities of one-dimensional linear stochastic differential equations. The quality of the approximation with respect to the noise amplitude is discussed and illustrated with numerical results for the spatially extended Hodgkin-Huxley equations. Python simulation code is supplied on GitHub under the link https://github.com/deristnochda/Hodgkin-Huxley-SPDE.

Fluctuation scaling in neural spike trains.

Fluctuation scaling has been observed universally in a wide variety of phenomena. In time series that describe sequences of events, fluctuation scaling is expressed as power function relationships between the mean and variance of either inter-event intervals or counting statistics, depending on measurement variables. In this article, fluctuation scaling has been formulated for a series of events in which scaling laws in the inter-event intervals and counting statistics were related. We have considered the first-passage time of an Ornstein-Uhlenbeck process and used a conductance-based neuron model with excitatory and inhibitory synaptic inputs to demonstrate the emergence of fluctuation scaling with various exponents, depending on the input regimes and the ratio between excitation and inhibition. Furthermore, we have discussed the possible implication of these results in the context of neural coding.

Developing software tools for parameter fitting and validation of neuronal models

Developing software tools for parameter fitting and validation of neuronal models

Channel-specific input/output transformations arising from the interaction between dynamic synapses and subthreshold oscillations

Subthreshold oscillations are observed in a wide variety of neurons in the nervous system (e.g. see [1,2]). They typically appear associated to resonance phenomena that allow, for instance, the implementation of intrinsic memory mechanisms for the detection of specific spike sequences in single neurons and neural networks [3-5]. On the other hand, synaptic transmission and, therefore, the corresponding input/output transformation in the postsynaptic cell are affected by recent presynaptic activity. Dynamic synapses modulate incoming information increasing (facilitation) or decreasing (depression) the postsynaptic response. This modulation provides additional characteristic time scales to the single neuron and network processing [6,7]. In the context of a neuron displaying subthreshold oscillations, the interplay between these additional time scales and the characteristic intrinsic fast and slow dynamics underlying the neuron's oscillatory activity can significantly affect its input/output transformation. Neuronal intrinsic subthreshold oscillations and dynamic synapses underlie several computational properties both at the single neuron and the network levels. Traditionally, intrinsic oscillations and dynamic synapses have been studied separately and their interaction has attracted almost no attention. In this work, we use a conductance-based neuron model and a dynamic synapse to investigate how intrinsic subthreshold oscillations and the short-term plasticity of a depressing synapse act together to shape its resonant properties and the corresponding input/output transformation. Our results suggest that factors such as the maximum hyperpolarization level, the oscillation amplitude and frequency or the resulting firing threshold can be modulated by synaptic depression. This shapes the postsynaptic neuron's resonant properties arising from the subthreshold oscillation and leads to complex channel-specific input/output relations. Thus, a low-cost modification in synaptic parameters can produce a significant different response. This complex synaptic-dependent input/output transformation allows the implementation of cost-effective information discrimination mechanisms in single neurons by just tuning the depression level of the synaptic channel without modifying the intrinsic neural dynamics.

Modeling the Influence of Ion Channels on Neuron Dynamics in Drosophila

Voltage gated ion channels play a major role in determining a neuron's firing behavior, resulting in the specific processing of synaptic input patterns. Drosophila and other invertebrates provide valuable model systems for investigating ion channel kinetics and their impact on firing properties. Despite the increasing importance of Drosophila as a model system, few computational models of its ion channel kinetics have been developed. In this study, experimentally observed biophysical properties of voltage gated ion channels from the fruitfly Drosophila melanogaster are used to develop a minimal, conductance based neuron model. We investigate the impact of the densities of these channels on the excitability of the model neuron. Changing the channel densities reproduces different in situ observed firing patterns and induces a switch from integrator to resonator properties. Further, we analyze the preference to input frequency and how it depends on the channel densities and the resulting bifurcation type the system undergoes. An extension to a three dimensional model demonstrates that the inactivation kinetics of the sodium channels play an important role, allowing for firing patterns with a delayed first spike and subsequent high frequency firing as often observed in invertebrates, without altering the kinetics of the delayed rectifier current.

Nonstationary filtered shot-noise processes and applications to neuronal membranes.

Filtered shot noise processes have proven to be very effective in modeling the evolution of systems exposed to shot noise sources and have been applied to a wide variety of fields ranging from electronics through biology. In particular, they can model the membrane potential V(m) of neurons driven by stochastic input, where these filtered processes are able to capture the nonstationary characteristics of V(m) fluctuations in response to presynaptic input with variable rate. In this paper we apply the general framework of Poisson point processes transformations to analyze these systems in the general case of nonstationary input rates. We obtain exact analytic expressions, as well as different approximations, for the joint cumulants of filtered shot noise processes with multiplicative noise. These general results are then applied to a model of neuronal membranes subject to conductance shot noise with a continuously variable rate of presynaptic spikes. We propose very effective approximations for the time evolution of the V(m) distribution and a simple method to estimate the presynaptic rate from a small number of V(m) traces. This work opens the perspective of obtaining analytic access to important statistical properties of conductance-based neuronal models such as the first passage time.

Analysis of the role of the low threshold currents IT and Ih in intrinsic delta oscillations of thalamocortical neurons.

Thalamocortical neurons are involved in the generation and maintenance of brain rhythms associated with global functional states. The repetitive burst firing of TC neurons at delta frequencies (1–4 Hz) has been linked to the oscillations recorded during deep sleep and during episodes of absence seizures. To get insight into the biophysical properties that are the basis for intrinsic delta oscillations in these neurons, we performed a bifurcation analysis of a minimal conductance-based thalamocortical neuron model including only the IT channel and the sodium and potassium leak channels. This analysis unveils the dynamics of repetitive burst firing of TC neurons, and describes how the interplay between the amplifying variable mT and the recovering variable hT of the calcium channel IT is sufficient to generate low threshold oscillations in the delta band. We also explored the role of the hyperpolarization activated cationic current Ih in this reduced model and determine that, albeit not required, Ih amplifies and stabilizes the oscillation.

Dendritic atrophy constricts functional maps in resonance and impedance properties of hippocampal model neurons.

A gradient in the density of hyperpolarization-activated cyclic-nucleotide gated (HCN) channels is necessary for the emergence of several functional maps within hippocampal pyramidal neurons. Here, we systematically analyzed the impact of dendritic atrophy on nine such functional maps, related to input resistance and local/transfer impedance properties, using conductance-based models of hippocampal pyramidal neurons. We introduced progressive dendritic atrophy in a CA1 pyramidal neuron reconstruction through a pruning algorithm, measured all functional maps in each pruned reconstruction, and arrived at functional forms for the dependence of underlying measurements on dendritic length. We found that, across frequencies, atrophied neurons responded with higher efficiency to incoming inputs, and the transfer of signals across the dendritic tree was more effective in an atrophied reconstruction. Importantly, despite the presence of identical HCN-channel density gradients, spatial gradients in input resistance, local/transfer resonance frequencies and impedance profiles were significantly constricted in reconstructions with dendritic atrophy, where these physiological measurements across dendritic locations converged to similar values. These results revealed that, in atrophied dendritic structures, the presence of an ion channel density gradient alone was insufficient to sustain homologous functional maps along the same neuronal topograph. We assessed the biophysical basis for these conclusions and found that this atrophy-induced constriction of functional maps was mediated by an enhanced spatial spread of the influence of an HCN-channel cluster in atrophied trees. These results demonstrated that the influence fields of ion channel conductances need to be localized for channel gradients to express themselves as homologous functional maps, suggesting that ion channel gradients are necessary but not sufficient for the emergence of functional maps within single neurons.

Spinal mechanisms may provide a combination of intermittent and continuous control of human posture: predictions from a biologically based neuromusculoskeletal model.

Several models have been employed to study human postural control during upright quiet stance. Most have adopted an inverted pendulum approximation to the standing human and theoretical models to account for the neural feedback necessary to keep balance. The present study adds to the previous efforts in focusing more closely on modelling the physiological mechanisms of important elements associated with the control of human posture. This paper studies neuromuscular mechanisms behind upright stance control by means of a biologically based large-scale neuromusculoskeletal (NMS) model. It encompasses: i) conductance-based spinal neuron models (motor neurons and interneurons); ii) muscle proprioceptor models (spindle and Golgi tendon organ) providing sensory afferent feedback; iii) Hill-type muscle models of the leg plantar and dorsiflexors; and iv) an inverted pendulum model for the body biomechanics during upright stance. The motor neuron pools are driven by stochastic spike trains. Simulation results showed that the neuromechanical outputs generated by the NMS model resemble experimental data from subjects standing on a stable surface. Interesting findings were that: i) an intermittent pattern of muscle activation emerged from this posture control model for two of the leg muscles (Medial and Lateral Gastrocnemius); and ii) the Soleus muscle was mostly activated in a continuous manner. These results suggest that the spinal cord anatomy and neurophysiology (e.g., motor unit types, synaptic connectivities, ordered recruitment), along with the modulation of afferent activity, may account for the mixture of intermittent and continuous control that has been a subject of debate in recent studies on postural control. Another finding was the occurrence of the so-called “paradoxical” behaviour of muscle fibre lengths as a function of postural sway. The simulations confirmed previous conjectures that reciprocal inhibition is possibly contributing to this effect, but on the other hand showed that this effect may arise without any anticipatory neural control mechanism.

High-conductance states and A-type K+ channels are potential regulators of the conductance-current balance triggered by HCN channels.

An increase in the hyperpolarization-activated cyclic nucleotide-gated (HCN) channel conductance reduces input resistance, whereas the consequent increase in the inward h current depolarizes the membrane. This results in a delicate and unique conductance-current balance triggered by the expression of HCN channels. In this study, we employ experimentally constrained, morphologically realistic, conductance-based models of hippocampal neurons to explore certain aspects of this conductance-current balance. First, we found that the inclusion of an experimentally determined gradient in A-type K(+) conductance, but not in M-type K(+) conductance, tilts the HCN conductance-current balance heavily in favor of conductance, thereby exerting an overall restorative influence on neural excitability. Next, motivated by the well-established modulation of neuronal excitability by synaptically driven high-conductance states observed under in vivo conditions, we inserted thousands of excitatory and inhibitory synapses with different somatodendritic distributions. We measured the efficacy of HCN channels, independently and in conjunction with other channels, in altering resting membrane potential (RMP) and input resistance (Rin) when the neuron received randomized or rhythmic synaptic bombardments through variable numbers of synaptic inputs. We found that the impact of HCN channels on average RMP, Rin, firing frequency, and peak-to-peak voltage response was severely weakened under high-conductance states, with the impinging synaptic drive playing a dominant role in regulating these measurements. Our results suggest that the debate on the role of HCN channels in altering excitability should encompass physiological and pathophysiological neuronal states under in vivo conditions and the spatiotemporal interactions of HCN channels with other channels.

Automatic parameter estimation of multicompartmental neuron models via minimization of trace error with control adjustment.

We describe a new technique to fit conductance-based neuron models to intracellular voltage traces from isolated biological neurons. The biological neurons are recorded in current-clamp with pink (1/f) noise injected to perturb the activity of the neuron. The new algorithm finds a set of parameters that allows a multicompartmental model neuron to match the recorded voltage trace. Attempting to match a recorded voltage trace directly has a well-known problem: mismatch in the timing of action potentials between biological and model neuron is inevitable and results in poor phenomenological match between the model and data. Our approach avoids this by applying a weak control adjustment to the model to promote alignment during the fitting procedure. This approach is closely related to the control theoretic concept of a Luenberger observer. We tested this approach on synthetic data and on data recorded from an anterior gastric receptor neuron from the stomatogastric ganglion of the crab Cancer borealis. To test the flexibility of this approach, the synthetic data were constructed with conductance models that were different from the ones used in the fitting model. For both synthetic and biological data, the resultant models had good spike-timing accuracy.

Evolutionary algorithm search for connectivity patterns conducive to bursting in respiratory neural networks

The respiratory neural network in the pre-Botzinger complex of the ventrolateral medulla controls and flexibly maintains the breathing rhythm, coordinating network-wide bursting to signal the inspiratory phase of the breath. Interestingly, however, the connectivity of the network by which it achieves this coordination is still a subject of active research. We search for connectivity patterns which yield high performance in a measure of network bursting, using an evolutionary algorithm (EA)-based approach, in networks of leaky integrate-and-fire (LIF) and conductance-based model (CBM) neurons. In LIF networks, the connectivity patterns of bursting networks are characterized by regularities in several network statistics: closeness centrality(CC), betweenness centrality (BC) and out-degree (OD) distributions. We find additionally that intentionally selecting for these statistical regularities in CC, BC and OD leads to steady improvement in the network bursting measure. We examine the extent to which these trends in network statistics are present in well-adapted networks of CBM neurons without intrinsically bursting neurons, finding that the regularities in BC and OD are also visible. We also present initial results on the statistical regularities in networks of CBM neurons with intrinsic bursters. Figure 1 illustrates the regularities in closeness centrality (CC), betweenness centality (BC) and out-degree (OD) distributions in well-adapted networks of LIF neurons. In well-adapted networks with (a) 2, (b) 3, (c) 4 and (d) 5 tonically spiking neurons present ...

A system for automated analysis of conductance correlations involved in recovery of electrical activity after neuromodulator deprivation in stomatogastric neuron models

Coregulation of ionic current levels, and specifically the changes that appear to take place in such relationships in response to deafferentation (i.e., neuromodulator deprivation), is thought to be one of the main explanations behind the phenomenon of recovery of electrical activity exhibited by stomatogastric (STG) neurons subjected to deafferentation [3]. Here, we are proposing an automated software system that allows for analysis of conductance correlations involved in recovery of electrical activity after deafferentation in STG neuron models. The system utilizes multi-objective evolutionary algorithms (MOEA) to explore the parameter space of conductance-based neuronal models in search of those models that exhibit activity resembling that of neurons in presence of neuromodulation, despite being simulated without it [1]. The system considers such models to represent “recovered” neurons, and compares the correlations discovered in a population of those models to relationships exhibited by “control” model neurons (i.e., those simulated with neuromodulators present). After the MOEA-based model construction is finished, the system automatically generates appropriate scatter plots, quantifies the relationships, finds differences between the two populations of models, and calculates the statistical significance of those differences. As a case study, we apply our system to the analyses of models of two very important STG neurons: the anterior burster (AB) and pyloric dilator (PD), proposed in [2]. In addition to demonstrating the applicability of this approach, we discuss interesting insights into the phenomenon of function recovery that all seem to involve the delayed rectifier (Kd) current.

Estimating the biophysical properties of neurons with intracellular calcium dynamics

We investigate the dynamics of a conductance-based neuron model coupled to a model of intracellular calcium uptake and release by the endoplasmic reticulum. The intracellular calcium dynamics occur on a time scale that is orders of magnitude slower than voltage spiking behavior. Coupling these mechanisms sets the stage for the appearance of chaotic dynamics, which we observe within certain ranges of model parameter values. We then explore the question of whether one can, using observed voltage data alone, estimate the states and parameters of the voltage plus calcium (V+Ca) dynamics model. We find the answer is negative. Indeed, we show that voltage plus another observed quantity must be known to allow the estimation to be accurate. We show that observing both the voltage time course V(t) and the intracellular Ca time course will permit accurate estimation, and from the estimated model state, accurate prediction after observations are completed. This sets the stage for how one will be able to use a more detailed model of V+Ca dynamics in neuron activity in the analysis of experimental data on individual neurons as well as functional networks in which the nodes (neurons) have these biophysical properties.

Self-organization of a recurrent network under ongoing synaptic plasticity

We investigated the organization of a recurrent network under ongoing synaptic plasticity using a model of neural oscillators coupled by dynamic synapses. In this model, the coupling weights changed dynamically, depending on the timing between the oscillators. We determined the phase coupling function of the oscillator model, Γ(ϕ), using conductance-based neuron models. Furthermore, we examined the effects of the Fourier zero mode of Γ(ϕ), which has a critical role in the case of spike-time-dependent plasticity-organized recurrent networks. Heterogeneous layered clusters with different frequencies emerged from homogeneous populations as the Fourier zero mode increased. Our findings may provide new insights into the self-assembly mechanisms of neural networks related to synaptic plasticity.

Cell-intrinsic mechanisms of temperature compensation in a grasshopper sensory receptor neuron.

Changes in temperature affect biochemical reaction rates and, consequently, neural processing. The nervous systems of poikilothermic animals must have evolved mechanisms enabling them to retain their functionality under varying temperatures. Auditory receptor neurons of grasshoppers respond to sound in a surprisingly temperature-compensated manner: firing rates depend moderately on temperature, with average Q10 values around 1.5. Analysis of conductance-based neuron models reveals that temperature compensation of spike generation can be achieved solely relying on cell-intrinsic processes and despite a strong dependence of ion conductances on temperature. Remarkably, this type of temperature compensation need not come at an additional metabolic cost of spike generation. Firing rate-based information transfer is likely to increase with temperature and we derive predictions for an optimal temperature dependence of the tympanal transduction process fostering temperature compensation. The example of auditory receptor neurons demonstrates how neurons may exploit single-cell mechanisms to cope with multiple constraints in parallel.DOI: http://dx.doi.org/10.7554/eLife.02078.001.

Active Dendrites Regulate Spectral Selectivity in Location-Dependent Spike Initiation Dynamics of Hippocampal Model Neurons

How does the presence of plastic active dendrites in a pyramidal neuron alter its spike initiation dynamics? To answer this question, we measured the spike-triggered average (STA) from experimentally constrained, conductance-based hippocampal neuronal models of various morphological complexities. We transformed the STA computed from these models to the spectral and the spectrotemporal domains and found that the spike initiation dynamics exhibited temporally localized selectivity to a characteristic frequency. In the presence of the hyperpolarization-activated cyclic nucleotide-gated (HCN) channels, the STA characteristic frequency strongly correlated with the subthreshold resonance frequency in the theta frequency range. Increases in HCN channel density or in input variance increased the STA characteristic frequency and its selectivity strength. In the absence of HCN channels, the STA exhibited weak delta frequency selectivity and the characteristic frequency was related to the repolarization dynamics of the action potentials and the recovery kinetics of sodium channels from inactivation. Comparison of STA obtained with inputs at various dendritic locations revealed that nonspiking and spiking dendrites increased and reduced the spectrotemporal integration window of the STA with increasing distance from the soma as direct consequences of passive filtering and dendritic spike initiation, respectively. Finally, the presence of HCN channels set the STA characteristic frequency in the theta range across the somatodendritic arbor and specific STA measurements were strongly related to equivalent transfer-impedance-related measurements. Our results identify explicit roles for plastic active dendrites in neural coding and strongly recommend a dynamically reconfigurable multi-STA model to characterize location-dependent input feature selectivity in pyramidal neurons.

Multiscale Piecewise Deterministic Markov Process in infinite dimension: central limit theorem and Langevin approximation

In [A. Genadot and M. Thieullen, Averaging for a fully coupled piecewise-deterministic markov process in infinite dimensions. Adv. Appl. Probab. 44 (2012) 749–773], the authors addressed the question of averaging for a slow-fast Piecewise Deterministic Markov Process (PDMP) in infinite dimensions. In the present paper, we carry on and complete this work by the mathematical analysis of the fluctuations of the slow-fast system around the averaged limit. A central limit theorem is derived and the associated Langevin approximation is considered. The motivation for this work is the study of stochastic conductance based neuron models which describe the propagation of an action potential along a nerve fiber.

EFFECTS OF EXTREMELY LOW-FREQUENCY MAGNETIC FIELDS ON THE RESPONSE OF A CONDUCTANCE-BASED NEURON MODEL

To provide insights into the modulation of neuronal activity by extremely low-frequency (ELF) magnetic field (MF), we present a conductance-based neuron model and introduce ELF sinusoidal MF as an additive voltage input. By analyzing spike times and spiking frequency, it is observed that neuron with distinct spiking patterns exhibits different response properties in the presence of MF exposure. For tonic spiking neuron, the perturbations of MF exposure on spike times is maximized at the harmonics of neuronal intrinsic spiking frequency, while it is maximized at the harmonics of bursting frequency for burst spiking neuron. As MF intensity increases, the perturbations also increase. Compared with tonic spiking, bursting dynamics are less sensitive to the perturbations of ELF MF exposure. Further, ELF MF exposure is more prone to perturb neuronal spike times relative to spiking frequency. Our finding suggests that the resonance may be one of the neural mechanisms underlying the modulatory effects of the low-intensity ELF MFs on neuronal activities. The results highlight the impacts of ELF MFs exposure on neuronal activity from the single cell level, and demonstrate various factors including ELF MF properties and neuronal spiking characteristics could determine the outcome of exposure. These insights into the mechanism of MF exposure may be relevant for the design of multi-intensity magnetic stimulus protocols, and may even contribute to the interpretation of MF effects on the central nervous systems.

Exploring action potential initiation in neurons exposed to DC electric fields through dynamical analysis of conductance-based model

Noninvasive direct current (DC) electric stimulation of central nervous system is today a promising therapeutic option to alleviate the symptoms of a number of neurological disorders. Despite widespread use of this noninvasive brain modulation technique, a generalizable explanation of its biophysical basis has not been described which seriously restricts its application and development. This paper investigated the dynamical behaviors of Hodgkin’s three classes of neurons exposed to DC electric field based on a conductance-based neuron model. With phase plane and bifurcation analysis, the different responses of each class of neuron to the same stimulation are shown to derive from distinct spike initiating dynamics. Under the effects of negative DC electric field, class 1 neuron generates repetitive spike through a saddle-node on invariant circle (SNIC) bifurcation, while it ceases this repetitive behavior through a Hopf bifurcation; Class 2 neuron generates repetitive spike through a Hopf bifurcation, meanwhile it ceases this repetitive behavior also by a Hopf bifurcation; Class 3 neuron can generate single spike through a quasi-separatrix-crossing (QSC) at first, then it generates repetitive spike through a Hopf bifurcation, while it ceases this repetitive behavior through a SNIC bifurcation. Furthermore, three classes of neurons’ spiking frequency f–electric field E (f–E) curves all have parabolic shape. Our results highlight the effects of external DC electric field on neuronal activity from the biophysical modeling point of view. It can contribute to the application and development of noninvasive DC brain modulation technique.