A Unified Approach to Linking Experimental,Statistical and Computational Analysis of Spike Train Data

A fundamental issue in neuroscience is how to identify the multiple biophysical mechanisms through which neurons generate observed patterns of spiking activity. In previous work, we proposed a method for linking observed patterns of spiking activity to specific biophysical mechanisms based on a state space modeling framework and a sequential Monte Carlo, or particle filter, estimation algorithm. We have shown, in simulation, that this approach is able to identify a space of simple biophysical models that were consistent with observed spiking data (and included the model that generated the data), but have yet to demonstrate the application of the method to identify realistic currents from real spike train data. Here, we apply the particle filter to spiking data recorded from rat layer V cortical neurons, and correctly identify the dynamics of an slow, intrinsic current. The underlying intrinsic current is successfully identified in four distinct neurons, even though the cells exhibit two distinct classes of spiking activity: regular spiking and bursting. This approach – linking statistical, computational, and experimental neuroscience – provides an effective technique to constrain detailed biophysical models to specific mechanisms consistent with observed spike train data.     

Molecular and physiological evidence for functional gamma-aminobutyric acid (GABA)-C receptors in growth hormone-secreting cells

The neurotransmitter gamma-aminobutyric acid (GABA), released by hypothalamic neurons as well as by growth hormone- (GH) and adrenocorticotropin-producing cells, is a regulator of pituitary endocrine functions. Different classes of GABA receptors may be involved. In this study, we report that GH cells, isolated by laser microdissection from rat pituitary slices, possess the GABA-C receptor subunit rho2. We also demonstrate that in the GH adenoma cell line, GH3, GABA-C receptor subunits are not only expressed but also form functional channels. GABA-induced Cl- currents were recorded using the whole cell patch clamp technique; these currents were insensitive to bicuculline (a GABA-A antagonist) but could be induced by the GABA-C agonist cis-4-aminocrotonic acid. In contrast to typical GABA-C mediated currents in neurons, they quickly desensitized. Ca2+i recordings were also performed on GH3 cells. The application of either GABA or cis-4-aminocrotonic acid led to Ca2+ transients of similar amplitude, indicating that the activation of GABA-C receptors in GH3 cells may cause membrane depolarization, opening of voltage-gated Ca2+ channels, and a subsequent Ca2+ influx. Our results point at a role for GABA in pituitary GH cells and disclose an additional pathway to the one known via GABA-B receptors.     

Biophysical basis for three distinct dynamical mechanisms of action potential initiation

Transduction of graded synaptic input into trains of all-or-none action potentials (spikes) is a crucial step in neural coding. Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties. Despite widespread use of this classification scheme, a generalizable explanation of its biophysical basis has not been described. We recorded from spinal sensory neurons representing each class and reproduced their transduction properties in a minimal model. With phase plane and bifurcation analysis, each class of excitability was shown to derive from distinct spike initiating dynamics. Excitability could be converted between all three classes by varying single parameters; moreover, several parameters, when varied one at a time, had functionally equivalent effects on excitability. From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin's classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents. Class 1 excitability occurs through a saddle node on invariant circle bifurcation when net current at perithreshold potentials is inward (depolarizing) at steady state. Class 2 excitability occurs through a Hopf bifurcation when, despite net current being outward (hyperpolarizing) at steady state, spike initiation occurs because inward current activates faster than outward current. Class 3 excitability occurs through a quasi-separatrix crossing when fast-activating inward current overpowers slow-activating outward current during a stimulus transient, although slow-activating outward current dominates during constant stimulation. Experiments confirmed that different classes of spinal lamina I neurons express the subthreshold currents predicted by our simulations and, further, that those currents are necessary for the excitability in each cell class. Thus, our results demonstrate that all three classes of excitability arise from a continuum in the direction and magnitude of subthreshold currents. Through detailed analysis of the spike-initiating process, we have explained a fundamental link between biophysical properties and qualitative differences in how neurons encode sensory input.     

Lyotropic Ion Channel Current Model Compared with Ising Model

Trans-membrane currents in ligand-gated ion channels are calculated in a non-equilibrium, chemically open whole cell system. The model is lyotropic in the sense that dynamics and parameters such as ligand concentration for half-maximal response (scale of response), and threshold for firing in neurons, are nonlinear functions of the reactant concentrations. The derived total current fits recorded data significantly better than those derived from mass action, Ising, and other equilibrium type models, in which the derived response can be displaced from the assessed response by several orders in the ligand concentration. A comparison of the model obtained with an Ising-like model provides a methodology to obtain the non-equilibrium scaling dependence of Ising-like models on the reactant concentrations.     

Extracting Information from the Power Spectrum of Synaptic Noise

In cortical neurons, synaptic "noise" is caused by the nearly random release of thousands of synapses. Few methods are presently available to analyze synaptic noise and deduce properties of the underlying synaptic inputs. We focus here on the power spectral density (PSD) of several models of synaptic noise. We examine different classes of analytically solvable kinetic models for synaptic currents, such as the "delta kinetic models," which use Dirac delta functions to represent the activation of the ion channel. We first show that, for this class of kinetic models, one can obtain an analytic expression for the PSD of the total synaptic conductance and derive equivalent stochastic models with only a few variables. This yields a method for constraining models of synaptic currents by analyzing voltage-clamp recordings of synaptic noise. Second, we show that a similar approach can be followed for the PSD of the the membrane potential (Vm) through an effective-leak approximation. Third, we show that this approach is also valid for inputs distributed in dendrites. In this case, the frequency scaling of the Vm PSD is preserved, suggesting that this approach may be applied to intracellular recordings of real neurons. In conclusion, using simple mathematical tools, we show that Vm recordings can be used to constrain kinetic models of synaptic currents, as well as to estimate equivalent stochastic models. This approach, therefore, provides a direct link between intracellular recordings in vivo and the design of models consistent with the dynamics and spectral structure of synaptic noise.     

Small Modifications to Network Topology Can Induce Stochastic Bistable Spiking Dynamics in a Balanced Cortical Model

Directed random graph models frequently are used successfully in modeling the population dynamics of networks of cortical neurons connected by chemical synapses. Experimental results consistently reveal that neuronal network topology is complex, however, in the sense that it differs statistically from a random network, and differs for classes of neurons that are physiologically different. This suggests that complex network models whose subnetworks have distinct topological structure may be a useful, and more biologically realistic, alternative to random networks. Here we demonstrate that the balanced excitation and inhibition frequently observed in small cortical regions can transiently disappear in otherwise standard neuronal-scale models of fluctuation-driven dynamics, solely because the random network topology was replaced by a complex clustered one, whilst not changing the in-degree of any neurons. In this network, a small subset of cells whose inhibition comes only from outside their local cluster are the cause of bistable population dynamics, where different clusters of these cells irregularly switch back and forth from a sparsely firing state to a highly active state. Transitions to the highly active state occur when a cluster of these cells spikes sufficiently often to cause strong unbalanced positive feedback to each other. Transitions back to the sparsely firing state rely on occasional large fluctuations in the amount of non-local inhibition received. Neurons in the model are homogeneous in their intrinsic dynamics and in-degrees, but differ in the abundance of various directed feedback motifs in which they participate. Our findings suggest that (i) models and simulations should take into account complex structure that varies for neuron and synapse classes; (ii) differences in the dynamics of neurons with similar intrinsic properties may be caused by their membership in distinctive local networks; (iii) it is important to identify neurons that share physiological properties and location, but differ in their connectivity.     

Oscillations in Large-Scale Cortical Networks: Map-Based Model

We develop a new computationally efficient approach for the analysis of complex large-scale neurobiological networks. Its key element is the use of a new phenomenological model of a neuron capable of replicating important spike pattern characteristics and designed in the form of a system of difference equations (a map). We developed a set of map-based models that replicate spiking activity of cortical fast spiking, regular spiking and intrinsically bursting neurons. Interconnected with synaptic currents these model neurons demonstrated responses very similar to those found with Hodgkin-Huxley models and in experiments. We illustrate the efficacy of this approach in simulations of one- and two-dimensional cortical network models consisting of regular spiking neurons and fast spiking interneurons to model sleep and activated states of the thalamocortical system. Our study suggests that map-based models can be widely used for large-scale simulations and that such models are especially useful for tasks where the modeling of specific firing patterns of different cell classes is important.     

Ligand-gated ion channel currents in a nonstationary lyotropic model

Transmembrane currents in ligand-gated ion channels are calculated in a nonstationary, chemically open whole cell system or patch of a membrane. The model is lyotropic in the sense that dynamics, and parameters such as the ligand concentration for half-maximal response (scale of response), and threshold for firing, such as in neurons, become nonlinear functions of the reactant concentrations. The derived total currents fit recorded data significantly better than those derived from mass action, Ising, and other stationary type models, in which the derived response is often displaced from the assessed response by several orders in the ligand concentration. Also, the derived slope of response is in perfect agreement with the values assessed.     

Dynamical estimation of neuron and network properties I: variational methods

We present a method for using measurements of membrane voltage in individual neurons to estimate the parameters and states of the voltage-gated ion channels underlying the dynamics of the neuron's behavior. Short injections of a complex time-varying current provide sufficient data to determine the reversal potentials, maximal conductances, and kinetic parameters of a diverse range of channels, representing tens of unknown parameters and many gating variables in a model of the neuron's behavior. These estimates are used to predict the response of the model at times beyond the observation window. This method of [Formula: see text] extends to the general problem of determining model parameters and unobserved state variables from a sparse set of observations, and may be applicable to networks of neurons. We describe an exact formulation of the tasks in nonlinear data assimilation when one has noisy data, errors in the models, and incomplete information about the state of the system when observations commence. This is a high dimensional integral along the path of the model state through the observation window. In this article, a stationary path approximation to this integral, using a variational method, is described and tested employing data generated using neuronal models comprising several common channels with Hodgkin-Huxley dynamics. These numerical experiments reveal a number of practical considerations in designing stimulus currents and in determining model consistency. The tools explored here are computationally efficient and have paths to parallelization that should allow large individual neuron and network problems to be addressed.     

Efficient maximum likelihood estimation of kinetic rate constants from macroscopic currents

A new method is described that accurately estimates kinetic constants, conductance and number of ion channels from macroscopic currents. The method uses both the time course and the strength of correlations between different time points of macroscopic currents and utilizes the property of semiseparability of covariance matrix for computationally efficient estimation of current likelihood and its gradient. The number of calculation steps scales linearly with the number of channel states as opposed to the cubic dependence in a previously described method. Together with the likelihood gradient evaluation, which is almost independent of the number of model parameters, the new approach allows evaluation of kinetic models with very complex topologies. We demonstrate applicability of the method to analysis of synaptic currents by estimating accurately rate constants of a 7-state model used to simulate GABAergic macroscopic currents.     

Probing the dynamics of identified neurons with a data-driven modeling approach

In controlling animal behavior the nervous system has to perform within the operational limits set by the requirements of each specific behavior. The implications for the corresponding range of suitable network, single neuron, and ion channel properties have remained elusive. In this article we approach the question of how well-constrained properties of neuronal systems may be on the neuronal level. We used large data sets of the activity of isolated invertebrate identified cells and built an accurate conductance-based model for this cell type using customized automated parameter estimation techniques. By direct inspection of the data we found that the variability of the neurons is larger when they are isolated from the circuit than when in the intact system. Furthermore, the responses of the neurons to perturbations appear to be more consistent than their autonomous behavior under stationary conditions. In the developed model, the constraints on different parameters that enforce appropriate model dynamics vary widely from some very tightly controlled parameters to others that are almost arbitrary. The model also allows predictions for the effect of blocking selected ionic currents and to prove that the origin of irregular dynamics in the neuron model is proper chaoticity and that this chaoticity is typical in an appropriate sense. Our results indicate that data driven models are useful tools for the in-depth analysis of neuronal dynamics. The better consistency of responses to perturbations, in the real neurons as well as in the model, suggests a paradigm shift away from measuring autonomous dynamics alone towards protocols of controlled perturbations. Our predictions for the impact of channel blockers on the neuronal dynamics and the proof of chaoticity underscore the wide scope of our approach.     

The response of cortical neurons to in vivo-like input current: theory and experiment

The study of several aspects of the collective dynamics of interacting neurons can be highly simplified if one assumes that the statistics of the synaptic input is the same for a large population of similarly behaving neurons (mean field approach). In particular, under such an assumption, it is possible to determine and study all the equilibrium points of the network dynamics when the neuronal response to noisy, in vivo-like, synaptic currents is known. The response function can be computed analytically for simple integrate-and-fire neuron models and it can be measured directly in experiments in vitro. Here we review theoretical and experimental results about the neural response to noisy inputs with stationary statistics. These response functions are important to characterize the collective neural dynamics that are proposed to be the neural substrate of working memory, decision making and other cognitive functions. Applications to the case of time-varying inputs are reviewed in a companion paper (Giugliano et al. in Biol Cybern, 2008). We conclude that modified integrate-and-fire neuron models are good enough to reproduce faithfully many of the relevant dynamical aspects of the neuronal response measured in experiments on real neurons in vitro.     

BDNF mediates the effects of testosterone on the survival of new neurons in an adult brain

New neurons are incorporated into the high vocal center (HVC), a nucleus of the adult canary (Serinus canaria) brain that plays a critical role in the acquisition and production of learned song. Recruitment of new neurons in the HVC is seasonally regulated and depends upon testosterone levels. We show here that brain-derived neurotrophic factor (BDNF) is present in the HVC of adult males but is not detectable in that of females, though the HVC of both sexes has BDNF receptors (TrkB). Testosterone treatment increases the levels of BDNF protein in the female HVC, and BDNF infused into the HVC of adult females triples the number of new neurons. Infusion of a neutralizing antibody to BDNF blocks the testosterone-induced increase in new neurons. Our results demonstrate that BDNF is involved in the regulation of neuronal replacement in the adult canary brain and suggest that the effects of testosterone are mediated through BDNF.     

Alterations in the expression and gating of Drosophila sodium channels by mutations in the para gene

Mutations in the para gene specifically affect the expression of sodium currents in Drosophila. While 65% of wild-type embryonic neurons in culture express sodium currents, three distinct mutations in the para locus resulted in a decrease in the fraction of cells from which sodium currents could be recorded. This reduction was allele-dependent: macroscopic sodium currents were exhibited in 49% of the neurons in parats1 cultures, 35% in parats2, and only 2% in paraST76. Voltage-clamp experiments demonstrated that the parats2 mutation also affected the gating properties of sodium channels. These results provide convincing evidence that para, a gene recently shown to exhibit sequence similarity to vertebrate sodium channels alpha subunits, encodes functional sodium channels in Drosophila. The finding that one para allele (paraST76) can virtually eliminate the expression of sodium currents strongly argues that the para gene codes for the majority of sodium channels in cultured embryonic neurons.     

Hormone-induced changes in identified cell populations of the higher vocal center in male canaries

Male canaries revise their vocal repertoire every year. Early work indicated that the volume and neuron number of the song-control nucleus HVC (Higher Vocal Center) declined in late-summer/fall as birds added and deleted syllables from their repertoire, and increased in spring as the set of song syllables stabilized to a fixed number. Seasonal variation in serum testosterone levels suggested that these changes in brain and behavior were regulated by testosterone (T). However, although initial studies describing growth and regression of HVC used Nissl-staining to define its borders, recent experiments that have measured the distribution of identified populations of HVC cells (projection neurons, hormone target cells) suggest that there are no seasonal changes in HVC volume or neuron number. In order to clarify the role of T in the regulation of HVC morphology, we castrated male canaries, maintained them on short (fall-like) days, and treated them with either T, antisteroid drugs, or nothing. After 1 month of treatment, we used a double-labeling technique to characterize HVC projection neurons and androgen target cells. The results showed that hormonal manipulation influenced HVC volume, the density and size of HVC cells, and the absolute number and percentage of androgen target cells in HVC. Hormonal manipulation did not influence the absolute number of cells in HVC. Moreover, the distribution of projection neurons, androgen target cells, and the Nissl-defined borders of HVC were closely aligned in all experimental groups, indicating that exposure to T and/or its metabolites (estradiol and dihydrotestosterone) regulates the overall size of HVC by affecting the distributions of both projection neurons and androgen target cells. Analysis of double-labeling results suggests that T specifically influences both cell size and the ability to accumulate androgen among HVC neurons that project to the robust nucleus of the archistriatum (RA). The results of this study show that steroid hormones exert potent effects on HVC morphology in male canaries, but differences between our results and studies of seasonal males suggest there may be additional factors that can regulate HVC morphology. © 1993 John Wiley & Sons, Inc.     

Transient Responses to Rapid Changes in Mean and Variance in Spiking Models

The mean input and variance of the total synaptic input to a neuron can vary independently, suggesting two distinct information channels. Here we examine the impact of rapidly varying signals, delivered via these two information conduits, on the temporal dynamics of neuronal firing rate responses. We examine the responses of model neurons to step functions in either the mean or the variance of the input current. Our results show that the temporal dynamics governing response onset depends on the choice of model. Specifically, the existence of a hard threshold introduces an instantaneous component into the response onset of a leaky-integrate-and-fire model that is not present in other models studied here. Other response features, for example a decaying oscillatory approach to a new steady-state firing rate, appear to be more universal among neuronal models. The decay time constant of this approach is a power-law function of noise magnitude over a wide range of input parameters. Understanding how specific model properties underlie these response features is important for understanding how neurons will respond to rapidly varying signals, as the temporal dynamics of the response onset and response decay to new steady-state determine what range of signal frequencies a population of neurons can respond to and faithfully encode.     

Changes in densities and kinetics of delayed rectifier potassium channels during neuronal differentiation

Single-channel K+ currents were recorded from young and mature spinal neurons cultured from Xenopus embryos to examine the bases of the developmental increases in density and in rate of activation of the macroscopic voltage-dependent delayed rectifier K+ current (IKv). K+ channels of three conductance classes (integral of 80, 30, and 15 pS) are present at both ages, but only the intermediate and small conductance classes are voltage-dependent and thus underlie IKv. The increase in the density of IKv is due to increases in the numbers of intermediate and small channels per cell, but not to changes in their open probabilities. The increase in rate of activation of IKv results from a change in the activation kinetics of the intermediate channel class alone.     

Single-channel currents underlying glycinergic inhibitory postsynaptic responses in spinal neurons.

Single-channel properties of glycine receptors have been characterized so far only in cultured neurons. To characterize the glycine receptor channels in situ, we applied the patch-clamp technique to spinal neurons in slice preparations. Glycine-gated, single-channel currents were recorded in outside-out patches excised from spinal neurons. In the falling phase of glycinergic inhibitory synaptic currents, single-channel currents were resolved as discrete steps. In both cases, the glycine-gated channels showed similar multiple conductance levels. These results suggest that the receptor channel properties are indistinguishable in the synaptic and extrasynaptic sites. We conclude that multiple conductance states of a receptor channel are the native feature of the glycine receptor in situ.     

Phylodynamic Inference for Structured Epidemiological Models

Coalescent theory is routinely used to estimate past population dynamics and demographic parameters from genealogies. While early work in coalescent theory only considered simple demographic models, advances in theory have allowed for increasingly complex demographic scenarios to be considered. The success of this approach has lead to coalescent-based inference methods being applied to populations with rapidly changing population dynamics, including pathogens like RNA viruses. However, fitting epidemiological models to genealogies via coalescent models remains a challenging task, because pathogen populations often exhibit complex, nonlinear dynamics and are structured by multiple factors. Moreover, it often becomes necessary to consider stochastic variation in population dynamics when fitting such complex models to real data. Using recently developed structured coalescent models that accommodate complex population dynamics and population structure, we develop a statistical framework for fitting stochastic epidemiological models to genealogies. By combining particle filtering methods with Bayesian Markov chain Monte Carlo methods, we are able to fit a wide class of stochastic, nonlinear epidemiological models with different forms of population structure to genealogies. We demonstrate our framework using two structured epidemiological models: a model with disease progression between multiple stages of infection and a two-population model reflecting spatial structure. We apply the multi-stage model to HIV genealogies and show that the proposed method can be used to estimate the stage-specific transmission rates and prevalence of HIV. Finally, using the two-population model we explore how much information about population structure is contained in genealogies and what sample sizes are necessary to reliably infer parameters like migration rates.     

Differential expression of voltage-gated calcium channels in identified visual cortical neurons.

Using the whole-cell patch-clamp technique, Ca2+ channel currents were examined in three distinct types of neurons derived from rat primary visual cortex. Callosal-projecting and superior colliculus-projecting neurons were identified following in vivo retrograde labeling with fluorescent "beads." A subset of intrinsic GABAergic visual cortical neurons was identified with the monoclonal antibody VC1.1. Although high voltage-activated Ca2+ channel currents were measured in all three cell types, clear differences in the densities of these channels were observed. There were also marked variations in the relative amplitudes of the inactivating and noninactivating components of the high voltage-activated currents, suggesting that N- and L-type Ca2+ channels are differentially distributed. Although low voltage-activated or T-type currents were measured in subsets of both types of projection neurons, they were not observed in VC1.1-positive cells. These results provide a direct demonstration that voltage-gated Ca2+ channels are expressed in neurons of the mammalian visual cortex and reveal that the distribution and densities of different Ca2+ channel types in diverse classes of visual cortical neurons are distinct.