Parameter Estimation Methods for Single Neuron Models

With the advancement in computer technology, it has become possible to fit complex models to neuronal data. In this work, we test how two methods can estimate parameters of simple neuron models (passive soma) to more complex ones (neuron with one dendritic cylinder and two active conductances). The first method uses classical voltage traces resulting from current pulses injection (time domain), while the second uses measures of the neuron's response to sinusoidal stimuli (frequency domain). Both methods estimate correctly the parameters in all cases studied. However, the time-domain method is slower and more prone to estimation errors in the cable parameters than the frequency-domain method. Because with noisy data the goodness of fit does not distinguish between different solutions, we suggest that running the estimation procedure a large number of times might help find a good solution and can provide information about the interactions between parameters. Also, because the formulation used for the model's response in the frequency domain is analytical, one can derive a local sensitivity analysis for each parameter. This analysis indicates how well a parameter is likely to be estimated and helps choose an optimal stimulation protocol. Finally, the tests suggest a strategy for fitting single-cell models using the two methods examined.     

Reduction of stochastic conductance-based neuron models with time-scales separation

We introduce a method for systematically reducing the dimension of biophysically realistic neuron models with stochastic ion channels exploiting time-scales separation. Based on a combination of singular perturbation methods for kinetic Markov schemes with some recent mathematical developments of the averaging method, the techniques are general and applicable to a large class of models. As an example, we derive and analyze reductions of different stochastic versions of the Hodgkin Huxley (HH) model, leading to distinct reduced models. The bifurcation analysis of one of the reduced models with the number of channels as a parameter provides new insights into some features of noisy discharge patterns, such as the bimodality of interspike intervals distribution. Our analysis of the stochastic HH model shows that, besides being a method to reduce the number of variables of neuronal models, our reduction scheme is a powerful method for gaining understanding on the impact of fluctuations due to finite size effects on the dynamics of slow fast systems. Our analysis of the reduced model reveals that decreasing the number of sodium channels in the HH model leads to a transition in the dynamics reminiscent of the Hopf bifurcation and that this transition accounts for changes in characteristics of the spike train generated by the model. Finally, we also examine the impact of these results on neuronal coding, notably, reliability of discharge times and spike latency, showing that reducing the number of channels can enhance discharge time reliability in response to weak inputs and that this phenomenon can be accounted for through the analysis of the reduced model.     

Spatially explicit models of Turelli-Hoffmann Wolbachia invasive wave fronts

This paper examines different mathematical models of insect dispersal and infection spread and compares these with field data. Reaction-diffusion and integro-difference equation models are used to model the spatio-temporal spread of Wolbachia in Drosophila simulans populations. The models include cytoplasmic incompatibility between infected females and uninfected males that creates a threshold density, similar to an Allee effect, preventing increase from low incidence of infection in the host population. The model builds on an earlier model (Turelli & Hoffmann, 1991) by incorporating imperfect maternal transmission. The results of simulations of the models using the same parameter values produce different dynamics for each model. These differences become very marked in the integro-difference equation models when insect dispersal patterns are assumed to be non-Gaussian. The success or failure of invasion by Wolbachia in the simulations may be attributed to the insect dispersal mechanism used in the model rather than the parameter values. As the models predict very different outcomes for the integro-difference models depending on the underlying assumptions of insect dispersal patterns, this emphasizes that good field data on real (rather than idealized) dispersal patterns need to be collected before models such as these can be used for predictive purposes.     

A self-organizing state-space-model approach for parameter estimation in hodgkin-huxley-type models of single neurons

Traditional approaches to the problem of parameter estimation in biophysical models of neurons and neural networks usually adopt a global search algorithm (for example, an evolutionary algorithm), often in combination with a local search method (such as gradient descent) in order to minimize the value of a cost function, which measures the discrepancy between various features of the available experimental data and model output. In this study, we approach the problem of parameter estimation in conductance-based models of single neurons from a different perspective. By adopting a hidden-dynamical-systems formalism, we expressed parameter estimation as an inference problem in these systems, which can then be tackled using a range of well-established statistical inference methods. The particular method we used was Kitagawa's self-organizing state-space model, which was applied on a number of Hodgkin-Huxley-type models using simulated or actual electrophysiological data. We showed that the algorithm can be used to estimate a large number of parameters, including maximal conductances, reversal potentials, kinetics of ionic currents, measurement and intrinsic noise, based on low-dimensional experimental data and sufficiently informative priors in the form of pre-defined constraints imposed on model parameters. The algorithm remained operational even when very noisy experimental data were used. Importantly, by combining the self-organizing state-space model with an adaptive sampling algorithm akin to the Covariance Matrix Adaptation Evolution Strategy, we achieved a significant reduction in the variance of parameter estimates. The algorithm did not require the explicit formulation of a cost function and it was straightforward to apply on compartmental models and multiple data sets. Overall, the proposed methodology is particularly suitable for resolving high-dimensional inference problems based on noisy electrophysiological data and, therefore, a potentially useful tool in the construction of biophysical neuron models.     

Molecular variability elicits a tunable switch with discrete neuromodulatory response phenotypes

Conductance-based compartment modeling requires tuning of many parameters to fit the neuron model to target electrophysiological data. Automated parameter optimization via evolutionary algorithms (EAs) is a common approach to accomplish this task, using error functions to quantify differences between model and target. We present a three-stage EA optimization protocol for tuning ion channel conductances and kinetics in a generic neuron model with minimal manual intervention. We use the technique of Latin hypercube sampling in a new way, to choose weights for error functions automatically so that each function influences the parameter search to a similar degree. This protocol requires no specialized physiological data collection and is applicable to commonly-collected current clamp data and either single- or multi-objective optimization. We applied the protocol to two representative pyramidal neurons from layer 3 of the prefrontal cortex of rhesus monkeys, in which action potential firing rates are significantly higher in aged compared to young animals. Using an idealized dendritic topology and models with either 4 or 8 ion channels (10 or 23 free parameters respectively), we produced populations of parameter combinations fitting the target datasets in less than 80 hours of optimization each. Passive parameter differences between young and aged models were consistent with our prior results using simpler models and hand tuning. We analyzed parameter values among fits to a single neuron to facilitate refinement of the underlying model, and across fits to multiple neurons to show how our protocol will lead to predictions of parameter differences with aging in these neurons.     

A comparative analysis of multi-conductance neuronal models <Emphasis Type="Italic">in silico</Emphasis>

We demonstrate that a previously presented flexible silicon–neuron architecture can implement three disparate conductance-based neuron models with both fast and slow dynamics. By exploiting the real-time nature of this physical implementation, we mapped the model dynamics across a large region of parameter space. We also found that two of these dynamically different models represent points in a contiguous bursting space that spans between the two models. By systematically varying the model parameters, we also found that multiple, diverse trajectories in parameter space connected the two canonical bursting points. In addition, we found that the combination of parameter values keeps the neuron in the bursting region. These findings demonstrate the usefulness of the silicon–neuron architecture as a neural-modeling tool and illustrate its versatility as a platform for a multi-behavioral neuron that resembles its living analog.     

Fitting experimental data to models that use morphological data from public databases

Ideally detailed neuron models should make use of morphological and electrophysiological data from the same cell. However, this rarely happens. Typically a modeler will choose a cell morphology from a public database, assign standard values for R _a, C _m, and other parameters and then do the modeling study. The assumption is that the model will produce results representative of what might be obtained experimentally. To test this assumption we developed models of CA1 hippocampal pyramidal neurons using 4 different morphologies obtained from 3 public databases. The multiple run fitter in NEURON was used to fit parameter values in each of the 4 morphological models to match experimental data recorded from 19 CA1 pyramidal cells. Fits with fixed standard parameter values produced results that were generally not representative of our experimental data. However, when parameter values were allowed to vary, excellent fits were obtained in almost all cases, but the fitted parameter values were very different among the 4 reconstructions and did not match standard values. The differences in fitted values can be explained by very different diameters, total lengths, membrane areas and volumes among the reconstructed cells, reflecting either cell heterogeneity or issues with the reconstruction data. The fitted values compensated for these differences to make the database cells and experimental cells more similar electrotonically. We conclude that models using fully reconstructed morphologies need to be calibrated with experimental data (even when morphological and electrophysiological data come from the same cell), model results should be generated with multiple reconstructions, morphological and experimental cells should come from the same strain of animal at the same age, and blind use of standard parameter values in models that use reconstruction data may not produce representative experimental results. Action Editor: Steve Redman     

Post-Hoc Pattern-Oriented Testing and Tuning of an Existing Large Model: Lessons from the Field Vole

Pattern-oriented modeling (POM) is a general strategy for modeling complex systems. In POM, multiple patterns observed at different scales and hierarchical levels are used to optimize model structure, to test and select sub-models of key processes, and for calibration. So far, POM has been used for developing new models and for models of low to moderate complexity. It remains unclear, though, whether the basic idea of POM to utilize multiple patterns, could also be used to test and possibly develop existing and established models of high complexity. Here, we use POM to test, calibrate, and further develop an existing agent-based model of the field vole (Microtus agrestis), which was developed and tested within the ALMaSS framework. This framework is complex because it includes a high-resolution representation of the landscape and its dynamics, of the individual’s behavior, and of the interaction between landscape and individual behavior. Results of fitting to the range of patterns chosen were generally very good, but the procedure required to achieve this was long and complicated. To obtain good correspondence between model and the real world it was often necessary to model the real world environment closely. We therefore conclude that post-hoc POM is a useful and viable way to test a highly complex simulation model, but also warn against the dangers of over-fitting to real world patterns that lack details in their explanatory driving factors. To overcome some of these obstacles we suggest the adoption of open-science and open-source approaches to ecological simulation modeling.     

Hands-on parameter search for neural simulations by a MIDI-controller

Computational neuroscientists frequently encounter the challenge of parameter fitting--exploring a usually high dimensional variable space to find a parameter set that reproduces an experimental data set. One common approach is using automated search algorithms such as gradient descent or genetic algorithms. However, these approaches suffer several shortcomings related to their lack of understanding the underlying question, such as defining a suitable error function or getting stuck in local minima. Another widespread approach is manual parameter fitting using a keyboard or a mouse, evaluating different parameter sets following the users intuition. However, this process is often cumbersome and time-intensive. Here, we present a new method for manual parameter fitting. A MIDI controller provides input to the simulation software, where model parameters are then tuned according to the knob and slider positions on the device. The model is immediately updated on every parameter change, continuously plotting the latest results. Given reasonably short simulation times of less than one second, we find this method to be highly efficient in quickly determining good parameter sets. Our approach bears a close resemblance to tuning the sound of an analog synthesizer, giving the user a very good intuition of the problem at hand, such as immediate feedback if and how results are affected by specific parameter changes. In addition to be used in research, our approach should be an ideal teaching tool, allowing students to interactively explore complex models such as Hodgkin-Huxley or dynamical systems.     

A novel connectionist framework for computation of an approximate convex-hull of a set of planar points, circles and ellipses

We propose a two layer neural network for computation of an approximate convex-hull of a set of points or a set of circles/ellipses of different sizes. The algorithm is based on a very elegant concept - shrinking of a rubber band surrounding the set of planar objects. Logically, a set of neurons is placed on a circle (rubber band) surrounding the objects. Each neuron has a parameter vector associated with it. This may be viewed as the current position of the neuron. The given set of points/objects exerts a force of attraction on every neuron, which determines how its current position will be updated (as if, the force determines the direction of movement of the neuron lying on the rubber band). As the network evolves, the neurons (parameter vectors) approximate the convex-hull more and more accurately. The scheme can be applied to find the convex-hull of a planar set of circles or ellipses or a mixture of the two. Some properties related to the evolution of the algorithm are also presented.     

How accurate are population models? Lessons from landscape‐scale tests in a fragmented system

There is a growing debate about the ability of Population Viability Analysis (PVA) to predict the risk of extinction. Previously, the debate has focused largely on models where spatial variation and species movement are ignored. We present a synthesis of the key results for an array of different species for which detailed tests of the accuracy of PVA models were completed. These models included spatial variation in habitat quality and the movement of individuals across a landscape. The models were good approximations for some species, but poor for others. Predictive ability was limited by complex processes typically overlooked in spatial population models, these being interactions between landscape structure and life history attributes. Accuracy of models could not be determined a priori, although model tests indicated how they might be improved. Importantly, model predictions were poor for some species that are among the best‐studied vertebrates in Australia. This indicated that although the availability of good life history data is a key part of PVA other factors also influence model accuracy. We were also able to draw broad conclusions about the sorts of populations and life history characteristics where model predictions are likely to be less accurate. Predictions of extinction risk are often essential for real‐world population management. Therefore, we believe that although PVA has been shown to be less than perfect, it remains a useful tool particularly in the absence of alternative approaches. Hence, tests of PVA models should be motivated by the cycle of testing and improvement.     

A model of a thalamic neuron

We modify our recent three equilibrium-point model of neuronal bursting by a means of a small deformation of the nullclines in the x-y phase plane to give a model that can have as many as five equilibrium points. In this model the middle stable equilibrium point (e.p.) is separated from the outer stable and unstable e.ps by two saddle points. If the system is started at rest at the middle stable e.p. it has the following complex properties: A short suprathreshold current pulse switches the model from a silent state to a bursting state, or to give a single burst, depending on the choice of parameters. A subthreshold depolarizing current step gives a passive response at rest, but if the model is either constantly hyperpolarized or constantly depolarized, then the same current step gives different active responses. At a hyperpolarized level this consists of a burst response that shows refractoriness. At a depolarized level it consists of tonic firing with a linear frequency--current relationship. Hyperpolarization from rest is followed by post-inhibitory rebound. The model responds in a unique and characteristic way to an applied current ramp. These properties are very similar to those that have been recently recorded intracellularly from neurons in the mammalian thalamus. In the x-y phase plane our models of the repetitively firing neuron, the bursting neuron and the thalamic neuron form a progression of models in which the y nullcline in the subthreshold region is deformed once to give the burst neuron model, and a second time to give the thalamic neuron model. Each deformation can be interpreted as corresponding to the inclusion of a slow inward current in the model. As these currents are included so the associated firing properties increase in complexity.     

景观破碎化对植物种群的影响

景观破碎化是目前存在的一种普遍现象,是由于人为因素或其它非人为因素的干扰所导致的景观破碎分离并由简单趋向复杂的过程。它直接或间接影响着景观的结构、功能及其动态。本文首先简要介绍景观破碎化的成因和不同研究角度与水平上景观破碎化影响的表现;然后着重分析景观破碎化对植物种群的大小和灭绝速率、扩散和迁入、遗传和变异以及存活力等的影响;同时归纳现阶段研究景观破碎化对植物种群影响的主要方法和模型;最后提出目前景观破碎化对植物种群影响研究中存在的三个主要问题：缺乏原始的资料、成熟的模型和破碎化与其他因素,如污染、气候变化等,交互作用的识别。     

不同尺度下城市景观综合指数的空间变异特征研究

在GIS与RS技术支持下,采用5 m分辨率的SPOT遥感图像数据,从城市土地利用角度,利用半变异函数对不同尺度的景观多样性、聚集度与分维数的空间变异进行了定量分析.结果表明,不同尺度下3种指数的空间变异具有相似特征,各个尺度上都具有空间依赖性,尺度越小,空间依赖性越大,空间变异的细节更显著,空间自相关性对总体变异的贡献逐渐增加,但尺度过小,有时会破坏景观内部结构.不同指数的半变异函数模型在相同尺度上差异显著,说明不同景观指数在不同尺度下的半变异函数模型不具可比性.就研究上海市内部土地利用结构而言,1 km的幅度是较合适的空间尺度.景观指数空间变异特征是尺度的函数,尺度对景观格局的影响不能忽视.景观综合指数对尺度响应的生态过程揭示了上海城市空间结构的规律性:在小尺度上的复杂无规律性,中尺度上的多中心性和大尺度上的圈层结构性,但各个尺度是相互依赖的,没有绝对界限.     

Dynamics and bifurcations of the adaptive exponential integrate-and-fire model

Recently, several two-dimensional spiking neuron models have been introduced, with the aim of reproducing the diversity of electrophysiological features displayed by real neurons while keeping a simple model, for simulation and analysis purposes. Among these models, the adaptive integrate-and-fire model is physiologically relevant in that its parameters can be easily related to physiological quantities. The interaction of the differential equations with the reset results in a rich and complex dynamical structure. We relate the subthreshold features of the model to the dynamical properties of the differential system and the spike patterns to the properties of a Poincaré map defined by the sequence of spikes. We find a complex bifurcation structure which has a direct interpretation in terms of spike trains. For some parameter values, spike patterns are chaotic.     

Intrinsic and extrinsic determinants of ion channel localization in neurons

Neurons are an extremely diverse group of excitable cells with a wide variety of morphologies including complex dendritic trees and very long axons. The electrical properties of neurons depend not only on the types of ion channels and receptors expressed, but also on where these channels are located in the cell. Two extreme examples that illustrate the subcellular polarized nature of neurons and the tight regulation of ion channel localization can be seen at the axon initial segment and the node of Ranvier. The axon initial segment is important for initiation of action potentials in the axon, whereas the node of Ranvier is required for the rapid, faithful and efficient propagation of action potentials along the axon. Given the similarity of their functions it is not surprising that nearly every protein component of the axon initial segment is also found at the node. However, there is one very important difference between these two sites: nodes require extrinsic, glial-derived factors in order to form, whereas the axon initial segment is intrinsically determined by the neuron. This mini-review discusses recent results that have begun to clarify the intrinsic and extrinsic mechanisms underlying formation of nodes and axon initial segments, and poses several important unanswered questions regarding their unique mechanisms of formation.     

Estimating parameters and predicting membrane voltages with conductance-based neuron models

Recent results demonstrate techniques for fully quantitative, statistical inference of the dynamics of individual neurons under the Hodgkin–Huxley framework of voltage-gated conductances. Using a variational approximation, this approach has been successfully applied to simulated data from model neurons. Here, we use this method to analyze a population of real neurons recorded in a slice preparation of the zebra finch forebrain nucleus HVC. Our results demonstrate that using only 1,500 ms of voltage recorded while injecting a complex current waveform, we can estimate the values of 12 state variables and 72 parameters in a dynamical model, such that the model accurately predicts the responses of the neuron to novel injected currents. A less complex model produced consistently worse predictions, indicating that the additional currents contribute significantly to the dynamics of these neurons. Preliminary results indicate some differences in the channel complement of the models for different classes of HVC neurons, which accords with expectations from the biology. Whereas the model for each cell is incomplete (representing only the somatic compartment, and likely to be missing classes of channels that the real neurons possess), our approach opens the possibility to investigate in modeling the plausibility of additional classes of channels the cell might possess, thus improving the models over time. These results provide an important foundational basis for building biologically realistic network models, such as the one in HVC that contributes to the process of song production and developmental vocal learning in songbirds.     

A small fraction of strongly cooperative sodium channels boosts neuronal encoding of high frequencies

Generation of action potentials (APs) is a crucial step in neuronal information processing. Existing biophysical models for AP generation almost universally assume that individual voltage-gated sodium channels operate statistically independently, and their avalanche-like opening that underlies AP generation is coordinated only through the transmembrane potential. However, biological ion channels of various types can exhibit strongly cooperative gating when clustered. Cooperative gating of sodium channels has been suggested to explain rapid onset dynamics and large threshold variability of APs in cortical neurons. It remains however unknown whether these characteristic properties of cortical APs can be reproduced if only a fraction of channels express cooperativity, and whether the presence of cooperative channels has an impact on encoding properties of neuronal populations. To address these questions we have constructed a conductance-based neuron model in which we continuously varied the size of a fraction [Formula: see text] of sodium channels expressing cooperativity and the strength of coupling between cooperative channels [Formula: see text]. We show that starting at a critical value of the coupling strength [Formula: see text], the activation curve of sodium channels develops a discontinuity at which opening of all coupled channels becomes an all-or-none event, leading to very rapid AP onsets. Models with a small fraction, [Formula: see text], of strongly cooperative channels generate APs with the most rapid onset dynamics. In this regime APs are triggered by simultaneous opening of the cooperative channel fraction and exhibit a pronounced biphasic waveform often observed in cortical neurons. We further show that presence of a small fraction of cooperative Na+ channels significantly improves the ability of neuronal populations to phase-lock their firing to high frequency input fluctuation. We conclude that presence of a small fraction of strongly coupled sodium channels can explain characteristic features of cortical APs and has a functional impact of enhancing the spike encoding of rapidly varying signals.     

On dependency properties of the ISIs generated by a two-compartmental neuronal model

One-dimensional leaky integrate and fire neuronal models describe interspike intervals (ISIs) of a neuron as a renewal process and disregarding the neuron geometry. Many multi-compartment models account for the geometrical features of the neuron but are too complex for their mathematical tractability. Leaky integrate and fire two-compartment models seem a good compromise between mathematical tractability and an improved realism. They indeed allow to relax the renewal hypothesis, typical of one-dimensional models, without introducing too strong mathematical difficulties. Here, we pursue the analysis of the two-compartment model studied by Lansky and Rodriguez (Phys D 132:267–286, 1999), aiming of introducing some specific mathematical results used together with simulation techniques. With the aid of these methods, we investigate dependency properties of ISIs for different values of the model parameters. We show that an increase of the input increases the strength of the dependence between successive ISIs.