Sequential estimation of intrinsic activity and synaptic input in single neurons by particle filtering with optimal importance density
- Pau Closas^{1}Email authorView ORCID ID profile and
- Antoni Guillamon^{2}
https://doi.org/10.1186/s13634-017-0499-3
© The Author(s) 2017
Received: 3 March 2017
Accepted: 18 August 2017
Published: 15 September 2017
Abstract
This paper deals with the problem of inferring the signals and parameters that cause neural activity to occur. The ultimate challenge being to unveil brain’s connectivity, here we focus on a microscopic vision of the problem, where single neurons (potentially connected to a network of peers) are at the core of our study. The sole observation available are noisy, sampled voltage traces obtained from intracellular recordings. We design algorithms and inference methods using the tools provided by stochastic filtering that allow a probabilistic interpretation and treatment of the problem. Using particle filtering, we are able to reconstruct traces of voltages and estimate the time course of auxiliary variables. By extending the algorithm, through PMCMC methodology, we are able to estimate hidden physiological parameters as well, like intrinsic conductances or reversal potentials. Last, but not least, the method is applied to estimate synaptic conductances arriving at a target cell, thus reconstructing the synaptic excitatory/inhibitory input traces. Notably, the performance of these estimations achieve the theoretical lower bounds even in spiking regimes.
Keywords
1 Introduction
Measurements of membrane potential traces constitute the main observable quantities to derive a biophysical neuron model. In particular, the dynamics of auxiliary variables and the model parameters are inferred from voltage traces, in a costly process that typically entails a variety of channel blocks and clamping techniques (see, for instance, [1]), as well as some uncertainty in the parameter values due to noise in the signal. Recent works in the literature deal with the problem of inferring hidden parameters of the model; see, for instance, [2–4] and, for an exhaustive review, [5]. In the same line, attempts to extract connectivity in networks of neurons from calcium imaging, see [6], are worth mentioning.
Apart from inferring intrinsic parameters of the model, voltage traces are also useful to obtain valuable information about synaptic input, an inverse problem with some satisfactory (see, for instance, [7–13]) but no complete solutions yet. The main shortcomings are the requirement of multiple (supposedly identical) trials for some methods to be applied and the need of avoiding signals obtained when ionic currents are active. The latter constraint arises from the fact that many methods rely on the linearity of the signal and this is not possible to achieve under quite general situations, like spiking regimes (see [14]) or subthreshold regimes when specific currents (e.g., AHP, LTS) are active (see [15]).
The problem investigated in this paper considers recordings of noisy voltage traces to infer the hidden gating variables of the neuron model, as well as filtered voltage estimates, model parameters, and input synaptic conductances.
- 1.
The time-evolving states characterizing the neuron dynamics, including a filtered membrane potential and the dynamics of the gating variables
- 2.
The parameters defining the neuron model
- 3.
The dynamics of synaptic conductances and its parameters, the final goal being to discern the temporal contributions of global excitation from those of global inhibition, g _{E}(t) and g _{I}(t), respectively.
An ideal method should be able to sequentially infer the time course of the membrane potential and its intrinsic/extrinsic activity from noisy observations of a voltage trace. The main features of the envisaged algorithm are fivefold: (i) Single-trial: the method should be able to estimate the desired signals and parameters from a single voltage trace, thus avoiding the experimental variability among trials; (ii) Sequential: the algorithm should provide estimates each time a new observation is recorded, thus avoiding re-processing of all data streams each time; (iii) Spike regime: contrary to most solutions operating only under the subthreshold assumption, the method should be able to operate in the presence of spikes as well; (iv) Robust: if the method is model-dependent, thus implying knowledge of the model parameters, then the algorithm should be provided with enhancements to adaptively learn these parameters; and (v) Statistically efficient: the performance of the method should be close to the theoretical lower bounds, meaning that the estimation error cannot be substantially reduced. Notice that the focus here is not on reducing the computational cost of the inference method, and thus, we allow ourselves to use resource-consuming algorithms. Indeed, the target application does not demand (at least as a main requirement) real-time operation, and thus, we prioritized performance (i.e., estimation accuracy and the rest of features described earlier) in our developments.
According to the above desired features, in this work, that substantially extends our previous contributions [16, 17], we are interested in methods that can provide estimations from a single trial and avoid the need of repetitions that could be contaminated by neuronal variability. Particularly, we concentrate on methods to extract intrinsic activity of ionic channels, namely the probabilities of opening and closing ionic channels, and the contribution of synaptic conductances. We built a method based on Bayesian theory to sequentially infer these quantities from single-trace, noisy membrane potentials. The material therein includes a discussion of the discrete state-space representation of the problem and the model inaccuracies due to mismodeling effects. We present two sequential inference algorithms: (i) a method based on particle filtering (PF) to estimate the time-evolving states of a neuron under the assumption of perfect model knowledge and (ii) an enhanced version where model parameters are jointly estimated, and thus, the rather strong assumption of perfect model knowledge is relaxed. We provide exhaustive computer simulation results to validate the algorithms and observe that they are attaining the theoretical lower bounds of accuracy, which are derived in the paper as well.
In this paper, we use the powerful tools of PF to make inferences in general state-space models. PF are a set of methods able to sample from the marginal filtering distribution in situations where analytical solutions are hard to work out or simply impossible. In the recent years, PFs played an important role in many research areas such as signal detection and demodulation, target tracking, positioning, Bayesian inference, audio processing, financial modeling, computer vision, robotics, control, or biology [18–24]. At a glance, PF approximates the filtering distribution of states given measurements by a set of random points, properly weighted according to Bayes’ rule. The generation of the random particles can be done through a variety of distributions, known as importance density. Particularly, we formulate the problem at hand and observe that it is possible to use the optimal importance density [25]. This distribution generates particles close to the target distribution, and thus, it can be shown to reduce the variance of the particles. As a consequence, for a fix number of particles, usage of this approach (not always possible) leads to better accuracy results than other choices. To the authors’ knowledge, the utilization of such sampling distribution is novel in the context of neural model filtering. Similar works have used PF to track neural dynamics, but with no optimal importance density (see [2, 3, 26]), or to estimate synaptic input from subthreshold recordings [11], as opposite to our proposed approach where we aim at providing estimates during the, highly nonlinear, spike regime. These references use the expectation-maximization algorithm to estimate the model parameters. In this paper, we use the Particle Marginal Markov-Chain Monte-Carlo (PMCMC) method to estimate the parameters. Lighter filtering methods based on the Gaussian assumption were considered in the literature (see [12, 13, 27] for instance), but the assumption might not hold in general, for instance, due to outliers in the membrane measurements or if more sophisticated models for the synaptic conductances are considered. In these situations, a PF approach seems more appropriate. As mentioned earlier, the focus here is on highly efficient and reliable filtering methods, rather than on computationally light inference methods. Summing up, our method jointly treats the features of handling single-voltage traces governed by nonlinear models, estimating both neuron parameters and synaptic conductances, even in the spiking regimes, and using optimal importance density for the PF together with a MCMC algorithm. The above references cope with some of these features, but to our knowledge, none of the recent methods in the literature accounts for all of them. It is worth noting that other simulation-based solutions can be adopted besides the PMCMC. For instance, the works [28–32] tackle state estimation and model fitting problems jointly.
The remainder of the article is organized as follows. In Section 2, we expose the problem and present the statistical model, essentially a discretization of the well-known Morris-Lecar model, and we analyze the model inaccuracies as well. Next, in Section 3, we present the different inference algorithms we apply depending on the knowledge of the system. Results are given in Section 4, where we tackle three inference problems: (i) when the parameters defining the model are known; (ii) when the parameters of the model are unknown, and thus, they need to be estimated; and (iii) estimation of synaptic conductances from voltage traces assuming unknown model parameters. Finally, Section 5 concludes the paper with final remarks.
2 Problem statement and model
2.1 Measurement modeling
- 1.
Voltage observations are noisy. This is due, in part, to the thermal noise at the sensing device, non-ideal conditions in experimental setups, etc.
- 2.
Recorded observations are discrete. All sensing devices record data by sampling at discrete time-instants k, at a sampling frequency f _{ s }=1/T_{ s }, the continuous-time natural phenomena. This is the task of an Analog-to-Digital Converter (ADC). Moreover, these samples are typically digitized, i.e., expressed by a finite number of bits. This latter issue is not tackled in the work as we assume that modern computer capabilities allow us to sample with relatively large number of bits per sample.
with v _{ k } representing the nominal membrane potential and \(\sigma _{y,k}^{2}\) modeling the noise variance due to the sensor or the instrumentation inaccuracies when performing the experiment. To provide comparable results, we define the signal-to-noise ratio (SNR) as SNR=P _{ s }/P _{ n }, with P _{ s } being the average signal power and \(P_{n}=\sigma _{y,k}^{2}\) the noise power.
2.2 Neural dynamical system
where u={E,I} and χ(t) is a zero-mean, white noise, Gaussian process with unit variance. Then, the OU process has mean g _{ u,0}, standard deviation σ _{ u }, and time constant τ _{ u }. This simple model was shown in [33] to yield a valid description of the synaptic noise, capturing the properties of more complex models. Other dynamics could be considered instead.
which is of the Markovian type.
with m _{ ∞ }(v _{ k }), n _{ ∞ }(v _{ k }) and τ _{ n }(v _{ k }) defined in Appendix 1. Then, (8) and (9) can be interpreted as x _{ k }=f _{ k }(x _{ k−1}).
2.3 State-space formulation
where the noise terms ν _{ v,k } and ν _{ n,k } are assumed jointly Gaussian with covariance matrix Σ _{ x,k }. Further details of this matrix are discussed in Section 2.4. The measurement noise ν _{ y,k } is assumed zero-mean, Gaussian, and with variance \(\sigma ^{2}_{y,k}\). Notice that the system is characterized by Gaussian distributions and is linear in the observations; this allows for an optimal design of the proposal density in the PF as exploited in Section 3.1.
This general form of Markovian type is preserved when the model is extended with a couple of OU processes associated to the excitatory and inhibitory synaptic conductances. In this case, the resulting state-space model is composed by the Morris-Lecar model used so far (with the peculiarity that the term −I _{syn} is added to (40)), plus the OU stochastic process in (3) describing I _{syn}. Therefore, the continuous-time state is x=(v,n,g _{E},g _{I})^{⊤}. The discrete version of the state-space is used again.
2.4 Model inaccuracies
The proposed estimation method relies on the fact that the neuron model is known. This is true to some extent, but most of the parameters in the Morris-Lecar model discussed are to be estimated. Typically, this model calibration is done beforehand, but as we will see later in Section 3.2, this could be done in parallel to the filtering process. Therefore, the robustness of the method to possible inaccuracies should be assessed. In this section, we point out possible causes of mismodeling. Computer simulations are later used to characterize the performance of the proposed methods under these impairments.
where I _{ o } is the nominal current applied and \(\sigma ^{2}_{I}\) the variance around this value. Plugging (12) into (8), we obtain that the contribution of I _{app} to the noise term is \(\frac {\mathrm {T}_{s}}{C_{m}}\nu _{I,k} \sim \mathcal {N} \left (0, (\mathrm {T}_{s}/C_{m})^{2}\sigma ^{2}_{I}\right)\).
where \(\bar {g}_{L}^{o}\) is the nominal, estimated conductance and \(\sigma ^{2}_{g}\) the variance of this estimate. Similarly, inserting (13) into (8), we see that the contribution of \(\bar {g}_{L}\) to the noise term is \(\frac {\mathrm {T}_{s}}{C_{m}}\nu _{g,k} \sim \mathcal {N} \left (0, (\mathrm {T}_{s}/C_{m})^{2}\left (v_{k-1}-E_{\mathrm {L}}\right)\sigma ^{2}_{g}\right)\).
Finally, the parameters in m _{ ∞ }(v _{ k }), n _{ ∞ }(v _{ k }), and τ _{ n }(v _{ k }) are to be estimated. In general, these parameters can be properly obtained off-line by standard methods; see [36]. However, as they are estimates, a residual error typically remains. To account for these inaccuracies, we consider that the equation governing the evolution of gating variables is corrupted by a zero-mean additive white Gaussian process with variance \(\sigma _{n}^{2}\).
as an estimate of \(\sigma _{v}^{2}\), provided accurate knowledge of \(\sigma ^{2}_{I}\) and \(\sigma ^{2}_{g}\). Otherwise, the covariance matrix of the process could be estimated by other means, as the ones presented in Section 3.2 for mixed state-parameter estimation in nonlinear filtering problems.
Notice that we are implicitly assuming white processes due to the diagonal structure of Σ _{ x,k }. It is worth mentioning that, should correlated noise be a more realistic model, the method proposed in this article would be still valid. The proposed method can cope with colored noise statistics since Σ _{ x,k } can be used seamlessly if it is not diagonal.
3 Methods
Two filtering methods are proposed here, depending on the knowledge regarding the underlying dynamical model. Section 3.1 presents an algorithm able to estimate the states in x _{ k } by a PF methodology, the particularity being that an optimal distribution is used to draw the random samples characterizing the joint filtering distribution of interest. This method assumes knowledge of the parameter values of the system model, although we account for some inaccuracies as detailed in Section 2.4. An enhanced version of this method is presented in Section 3.2, where we relax the assumption of knowing the parameter values. Leveraging on a PMCMC algorithm and the use of the optimal importance density as in the first method, we present a method that is able to filter x _{ k } while estimating the values describing the neuron model.
3.1 Sequential estimation of voltage traces and gating variables
The type of problems we are interested in involve the estimation of time-evolving signals that can be expressed through a state-space formulation. Particularly, estimation of the states in a single-neuron model from noisy voltage traces can be readily seen as a filtering problem. Bayesian theory provides the mathematical tools to deal with such problems in a systematic manner. The focus is on sequential methods that can incorporate new available measurements as they are recorded without the need for reprocessing all past data.
with p(y _{ k }|x _{ k }) and p(x _{ k }|x _{ k−1}) referred to as the likelihood and the prior distributions, respectively. Unfortunately, (16) can only be obtained in closed-form in some special cases, and in more general setups, we should resort to more sophisticated methods. In this paper, we consider PF to cope with the nonlinearity of the model. Although other lighter approaches might be possible as well [22], we seek the maximum accuracy regardless the involved computational cost. Theoretically, for sufficiently large number of particles, particle filters offer such performances.
where \(\hat {{\boldsymbol {x}}}_{k} = \left (\hat {v}_{k}, \hat {n}_{k} \right)^{\top }\). Recall that the method discussed in this section could be easily adapted to other neuron models by simply substituting the corresponding transition function f _{ k } and constructing the state vector x _{ k } accordingly.
When we add the dynamics of the synaptic conductances, the vector Θ of model parameters also includes τ _{ E }, τ _{ I }, g _{ E,0}, g _{ I,0}, σ _{ E }, and σ _{ I }.
3.2 Joint estimation of states and model parameters
In practice, the parameters in (24) might not be known. It is reasonable to assume that Θ, or a subset of these parameters θ⊆Θ, are unknown and need to be estimated at the same time the filtering method in Algorithm 1 is executed. Therefore, the ultimate goal in this case is to estimate jointly the time evolving states and the unknown parameters of the model, x _{1:k } and θ, respectively.
Joint estimation of states and parameters is a longstanding problem in Bayesian filtering and specially hard to handle in the context of PFs. Refer to [38–40] and their references for a complete survey. Here, we follow the approach in [41] and make use of the so-called PMCMC to enhance the presented PF algorithm with parameter estimation capabilities. In the remainder of this section, we provide the basic ideas to use the algorithm, following a similar approach as in [24]. Connections to other methods are discussed in [42].
for k≥1.
Then, the basic problem is to obtain an estimate of the predictive distribution p(y _{ k }|y _{1:k−1},θ) from the PF we have designed in Section 3.1 and use it in conjunction with p(θ) to infer the marginal distribution p(θ|y _{1:T }) of interest. This latter step can be performed in several ways, from which we choose to use the Markov-Chain Monte-Carlo (MCMC) methodology to continue with a fully Bayesian solution. Besides, it is known to be the solution that provides the best results when used in a PF. Next, we detail how φ _{ k }(θ) can be obtained from a PF algorithm, we present the MCMC method for parameter estimation, and finally we sketch the overall algorithm.
3.2.1 Computing the energy function from particle filters
which can be computed recursively in the PF algorithm and becomes an approximation \( \hat {\varphi }_{T}(\boldsymbol {\theta })\) of the energy function.
3.2.2 The Particle Markov-Chain Monte-Carlo algorithm
is referred to as the acceptance probability.
The main assumption in Algorithm 2 is the ability of evaluating the energy function, φ _{ T }(·). We have seen earlier how this can be done in a PF. Roughly speaking, the PMCMC algorithm consists of putting together these two algorithms [41]. We refer to Algorithm 3 for the resulting PMCMC method.
4 Computer simulation results
We simulated the data of a neuron following the Morris-Lecar model. Particularly, we generated data sampled at f _{ s }=4 kHz. Notice that typical sampling rates are on the order of kilohertz, therefore ensuring that we are operating in the regime where the Nyquist rate is well satisfied (that is, f _{ s }>2·BW, with BW the bandwidth of the recorded signal) [1, Chapter 3]. The model parameters were set to C _{ m }=20 μF/cm^{2}, ϕ=0.04, V _{1}=−1.2 mV, V _{2}=18 mV, V _{3}=2 mV, and V _{4}=30 mV; the reverse potentials were E _{L}=−60 mV, E _{Ca}=120 mV, and E _{K}=−84 mV; and the maximal conductances were \(\bar {g}_{\text {Ca}}=4.4~\mathrm {mS/cm}^{2}\) and \(\bar {g}_{\mathrm {K}}=8.0~\mathrm {mS/cm}^{2}\). We considered a measurement noise with a standard deviation of 1 mV, which corresponds to an SNR of 32 dB. This value is considered a reasonable value in nowadays intracellular sensing devices. Model inaccuracies were generated as in Section 2.4.
Three sets of simulations are discussed. First, we validated the filtering method considering perfect knowledge of the model. In this case, the method in Algorithm 1 was used. Secondly, the model assumptions were relaxed in the sense that the parameters of the model were not known by the method. We analyzed the capabilities of the proposed method to infer both the time-evolving states of the system and some of the parameters defining the model. In this case, the method in Algorithm 3 was used. Finally, we validated the performance of the proposed methods in inferring the synaptic conductances. We tested both PF and PMCMC methods, that is, with and without full knowledge of the model, respectively.
4.1 Model parameters are known
In the simulations, we considered the model inaccuracies described in Section 2.4. To excite the neuron into spiking activity, a nominal applied current was injected with I _{ o }=110 μA/cm ^{2} and two values for σ _{ I } were considered, namely 1 and 10% of I _{ o }. The nominal conductance used in the model was \(\bar {g}_{\mathrm {L}}=2~\mathrm {mS/cm}^{2}\), whereas the underlying neuron had a zero-mean Gaussian error with standard deviation \(\sigma _{\bar {g}_{\mathrm {L}}}\). Two variance values were considered as well, 1 and 10% of \(\bar {g}_{\mathrm {L}}\). Finally, we considered σ _{ n }=10^{−3} in the dynamics of the gating variable.
where \(\hat {w}_{j,k}\) denotes the estimate of w _{ k } at the jth realization and M the number of independent Monte-Carlo trials used to approximate the mathematical expectation.
Averaged results over simulation time
σ _{ I }=0.01·I _{ o }, | σ _{ I }=0.1·I _{ o }, | |||
---|---|---|---|---|
\(\sigma _{g_{\mathrm {L}}} = 0.01 \cdot \bar {g}_{\mathrm {L}}^{o}\) | \(\sigma _{g_{\mathrm {L}}} = 0.1 \cdot \bar {g}_{\mathrm {L}}^{o}\) | |||
N=500 | N=1000 | N=500 | N=1000 | |
〈RMSE(v _{ k }) 〉 | 0.3344 | 0.3211 | 0.4269 | 0.4203 |
〈PCRB(v _{ k }) 〉 | 0.2325 | 0.2325 | 0.3777 | 0.3777 |
〈RMSE(n _{ k }) 〉 | 0.0046 | 0.0045 | 0.0056 | 0.0055 |
〈PCRB(n _{ k }) 〉 | 0.0043 | 0.0043 | 0.0053 | 0.0053 |
4.2 Model parameters are unknown
In this section, we validate the algorithm presented in Section 3.2. According to the previous analysis, we deem that 500 particles are enough for the filter to provide reliable results. The parameters of the PMCMC algorithm were set to γ=0.9 and \(\bar {\alpha }_{\ast } = 0.234\).
True value, initial value, and covariance of the parameters in Fig. 5
Parameter | True value | Initial value | Init. covariance |
---|---|---|---|
\(\bar {g}_{\text {Ca}}\) | 4.4 | 8 | 1 |
\(\bar {g}_{\mathrm {K}}\) | 8 | 5 | 1 |
σ _{ v } | 0.0307 | 0.05 | 0.01 |
σ _{ n } | 0.001 | 0.01 | 0.001 |
σ _{ y,k } | 1 | 10 | 0.5 |
It can be observed that the filtering performances with perfect knowledge of the model and with estimation of parameters by PMCMC are similar. Moreover, both approaches attain the theoretical lower bound of accuracy given by the PCRB, which is derived from the true model; see Appendix 2.
4.3 Estimation of synaptic conductances
Finally, once the methods to estimate state variables and unknown parameters were consolidated, we proceeded to test the methods to our ultimate goal: estimating jointly the intrinsic states of the neuron and the extrinsic inputs (i.e., the synaptic conductances).
True value, initial value, and covariance of the parameters in Fig. 9
Parameter | True value | Initial value | Init. covariance |
---|---|---|---|
τ _{E} | 2.73 | 1.5 | 1 |
g _{E,0} | 12.1 | 10 | 1 |
σ _{E} | 12 | 25 | 5 |
τ _{I} | 10.49 | 15 | 10 |
g _{I,0} | 57.3 | 45 | 10 |
σ _{I} | 26.4 | 35 | 5 |
We refer to the Additional file 1 to visualize a dynamic simulation showing how the estimations evolve as the PMCMC algorithm was applied in a case where the values of \(\bar {g}_{\mathrm {L}}\) and E _{L} were unknown.
Additional file 1: Evolution of the iterative learning when estimating leakage parameters. (MP4 2129 KB)
5 Conclusions
In this paper, we propose a filtering method that is able to sequentially infer the time course of the membrane potential, the intrinsic activity of ionic channels, and the input synaptic conductances from noisy observations of voltage traces. The method works both for subthreshold and spiking regimes. It is based on the PF methodology and features an optimal importance density, providing enhanced use of the particles that characterize the filtering distribution. In addition, we tackle the problem of joint parameter estimation and state filtering by extending the designed PF with an MCMC procedure in an iterative method known as PMCMC. Another distinctive contribution with respect to the other works in the literature is that here, we provide accuracy bounds for the problem at hand, given by the PCRB. The RMSEs of our methods are then compared to the bound, and therefore, we can assess the efficiency of the proposed inference methods.
Filtering methods of different types (e.g., PF or sigma-point Kalman filtering) have been used in other recent contributions to similar problems; see [7–13]. From a methodological perspective, the novelty of this paper is in the use of an optimal importance density to generate particles, a fact that increases the estimation accuracy for a given budget of particles. This technical detail only applies to PF methods. Although Gaussian methods (e.g., the family of sigma-point Kalman filters) have a lower computational cost in general, they require Gaussianity of the measures, whereas PFs do not. This is an advantage that we think can be crucial in estimating synaptic conductances, since the assumption of Gaussianity is generally assumed in the literature [4, 33], but there are no conclusive evidences to assert this assumption. In this paper, we have still applied the PF to an Ornstein-Uhlenbeck process in order to check that basic results can be attained, but further research will go in the direction of assuming other types of distributions for the synaptic conductances. The use of more complex distributions nicely fits within the framework of our PF-based method. Another advantage of PFs versus Gaussian filters is their enhanced robustness to outliers [53]; for instance, due to recording artifacts, future applications shall also incorporate this feature.
We have found excellent estimations of the synaptic conductances, even in spiking regimes. Estimating synaptic conductances in spiking regimes is a challenge which is far to be solved. It is well known that linear estimations of synaptic conductances are not trustable in this situation when data is extracted intracellularly from the spiking activity of neurons; see [14]. In experiments, thus, caution has to be taken in eliminating part of the voltage traces, thus losing also part of the temporal information of both excitatory and inhibitory conductances. Our method is able to perform well in this regime. This information is highly valuable in problems (epilepsy, schizophrenia, Alzheimer’s disease, etc.) where a debate on the balance of excitation and inhibition is open; see the introduction of [12] for a rather complete overview on this feature.
The results show the validity of the approach when applied to a Morris-Lecar type of neuron. However, the procedure is general and could be applied to any neuron model exhibiting more complex dynamics like bursting and mixed-mode oscillations. Nevertheless, a clear drawback is the need for specifying a model although its parameters are estimated by the method. This is an ubiquitous problem in other model-based methods. Future research includes the enhancement of the proposed method to account for model variability, for instance, the use of Interacting Multiple Model (IMM) approaches. Other forthcoming applications could be validating the method using real data recordings, both for inferring parameters of the model and synaptic conductances. The latter problem is a challenging hot topic in the neuroscience literature, which is recently focusing on methods to extract the conductances from single-trace measurements. We think that our PF method would give useful and interesting results to physiologists that aim at inferring the brain’s activation rules from neurons’ activities. Actually, knowing the excitatory-inhibitory time-course separation can help in getting important conclusions about the brain’s functional connectivity (see [54–56]).
We have not tried to obtain estimations when subthreshold ionic currents are active, where the presence of nonlinearities could also contaminate the estimations; see [15]. According to the excellent performance in spiking regimes, where nonlinearities are stronger, we expect also a good agreement between the estimated data and the prescribed synaptic conductances. Other extensions of the model can be devoted to incorporate the dendro-somatic interaction (see, for instance, [57]), by considering multi-compartmental neuron models, thus taking into account the morphology and the functional properties of the cell. This is another big challenge for which we think that our method can account for.
6 Appendix 1: Morris-Lecar neuron model
respectively. \(\bar {g}_{\mathrm {L}}\), \(\bar {g}_{\text {Ca}}\), and \(\bar {g}_{\mathrm {K}}\) are the maximal conductances of each current. E _{L}, E _{Ca}, and E _{K} denote the Nernst equilibrium potentials, for which the corresponding current is zero, a.k.a. reverse potentials.
which parameters V _{1}, V _{2}, V _{3}, and V _{4} can be measured experimentally [36].
The knowledgeable reader would have noticed that the Morris-Lecar model is a Hodgin-Huxley-type model with the usual considerations, where the following two extra assumptions were made: the depolarizing current is generated by Ca ^{2+} ionic channels (or Na ^{+} depending on the type of neuron modeled), whereas hyperpolarization is carried by K ^{+} ions, and that m=m _{ ∞ }(v). The Morris-Lecar model is very popular in computational neuroscience as it models a large variety of neural dynamics while its phase-plane analysis is more manageable as it involves only two states [35].
The Morris-Lecar, although simple to formulate, results in a very interesting model as it can produce a number of different dynamics. For instance, for given values of its parameters, we encounter a subcritical Hopf bifurcation for I _{app}=93.86 μA/cm ^{2}. On the other hand, for another set of parameter values, the system of equations has a Saddle-Node on an Invariant Circle (SNIC) bifurcation at I _{app}=39.96 μA/cm ^{2}.
7 Appendix 2: PCRB in Morris-Lecar models
where \(\hat {{\boldsymbol {x}}}_{k}(y_{1:k})\) represents an estimator of x _{ k } given y _{1:k }.
for some initial J _{0}. Notice that, in our case, \(\mathbf {D}_{k}^{22}\) becomes deterministic, but the rest of the terms involving expectations should be computed by Monte-Carlo integration over independent state trajectories.
Declarations
Funding
The work of the second author was supported by the project MTM2015-71509-C2-2-R (MINECO/FEDER) and by the grant 2014-SGR-504 (Government of Catalonia).
Authors’ contributions
Both authors contributed equally to this work. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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