Fokker–Planck and Fortet Equation-Based Parameter Estimation for a Leaky Integrate-and-Fire Model with Sinusoidal and Stochastic Forcing
© A. Iolov et al.; licensee Springer 2014
Received: 30 December 2012
Accepted: 6 June 2013
Published: 17 April 2014
Analysis of sinusoidal noisy leaky integrate-and-fire models and comparison with experimental data are important to understand the neural code and neural synchronization and rhythms. In this paper, we propose two methods to estimate input parameters using interspike interval data only. One is based on numerical solutions of the Fokker–Planck equation, and the other is based on an integral equation, which is fulfilled by the interspike interval probability density. This generalizes previous methods tailored to stationary data to the case of time-dependent input. The main contribution is a binning method to circumvent the problems of nonstationarity, and an easy-to-implement initializer for the numerical procedures. The methods are compared on simulated data.
List of Abbreviations
LIF: Leaky integrate-and-fire
ISI: Interspike interval
SDE: Stochastic differential equation
PDE: Partial differential equation
Information processing in the nervous system is carried out by spike timings in neurons. To study the neural code in such a complicated system, a first step is to understand signal processing and transmission in single neurons. Stochastic leaky integrate-and-fire (LIF) neuronal models are a good compromise between biophysical realism and mathematical tractability, and are commonly applied as theoretical tools to study properties of real neuronal systems. A central issue is then to perform statistical inference from experimental data and estimate model parameters. Many electrophysiological experiments on neurons, namely extra-cellular recordings, are only capable of detecting the time of the spike and not the detailed voltage trajectory leading up to the spike. Estimating the parameters of the LIF model from this type of data is equivalent to estimating the parameters of a stochastic model from the statistics of the first-passage times only. A common assumption is that the data are well described by a renewal process, thus basing the statistical inference on the interspike intervals (ISIs), assuming these are realizations of independent and identically distributed random variables. Since only partial information about the process is available, the statistical problem becomes more difficult, and no explicit expression for the likelihood is available.
Different methods have been proposed. In the seminal paper , a point process approach is proposed. The spike trains of a collection of neurons are represented as counting processes. Time is discretized and the point processes approximated by 0–1 time series. Then the probability of firing in the next time interval is modeled as a function of the spike history. In this way, maximum likelihood estimation is feasible. External stimuli are not considered. In , a numerically involved moment method is developed. It uses the first two moments of the first-passage times of the Ornstein–Uhlenbeck process to a constant threshold, which are given as series expressions, and equates them to their empirical counterparts. In [3, 4], certain explicit moment relations derived from the Laplace transform of the first-passage time distribution are applied, but these are only valid under stimulation (supra-threshold regime). In , inference is based on numerical inversion of the Laplace transform. In , a functional of a three-dimensional Bessel bridge is applied to obtain a maximum likelihood estimator. None of these methods are feasible to extend to the time-inhomogeneous case, which is of our interest. In [7, 8], an integral equation is used to derive an estimator in the time-homogeneous setting. This approach is readily extended to time varying input, which we will explore in this paper. Some of the above methods are compared in . Finally, a review of estimation methods is provided in .
Many sensory stimuli, like sound, contain an oscillatory component [11, 12]. Such inputs will cause oscillating membrane potentials in the neuron, generating rhythmic spiking patterns. The oscillation frequency determines the basic rhythm of spiking, and is considered to be significant for neuronal information processing. The dynamics of periodically forced neuron models have been extensively studied; see [13–18] and references therein. Even so, attempts to solve the estimation problem in these nonstationary settings have been rare. One problem is that the ISIs are no longer independent nor identically distributed. In , a more complicated model with linear filters is considered, allowing also for the spike history to influence the membrane potential dynamics. The estimation problem is solved through numerical solutions to the Fokker–Planck equation, and it is shown that the log-likelihood is concave, thus ensuring a global maximum; see also [20, 21]. Because their model is more involved, some approximations to the solution of the Fokker–Planck equation are applied, to ensure acceptable computing times. We will apply the full Fokker–Planck equation to solve our estimation problem, since the computing time is always lower than 2 seconds for a sample size of 1000 spikes.
In this paper, we thus describe and discuss two methods to estimate parameters of LIF models with the added complexity of a time-varying input current. We assume that the time-varying current is a sinusoidal wave, but we believe that the approaches generalize to an arbitrary periodic forcing with known frequency. One approach relies on the Fortet integral equation, which is readily extended to the time-inhomogeneous case. An advantage of this approach is that if the transition density of the diffusion in the LIF model is known, as is the case for the Ornstein–Uhlenbeck and the Feller model, the computational burden is limited. A second approach involves numerical solution of the Fokker–Planck equation, where the time-dependence is explicitly accounted for. After a numerical differentiation, the likelihood function can be calculated providing the maximum likelihood estimator. Nevertheless, we chose an alternative loss function which seems marginally more robust, directly comparing the survival function provided by the solution of the Fokker–Planck equation with its empirical counterpart. The two approaches give similar results and they are more carefully compared in the supplementary online material. Both methods need sensible starting values for the optimization algorithms, and we provide an easy-to-implement initializer. The estimation procedures are compared on simulated data and we find that both algorithms are able to find estimates close to the true values for several different dynamical regimes. We find that for small sample sizes the Fokker–Planck algorithm can be considered marginally preferable, whereas for larger sample sizes the Fortet algorithm becomes marginally superior. Moreover, at high frequencies of the sinusoidal forcing, the Fortet is better at identifying the parameters, though in general there is less information in the data to distinguish between a constant input and the amplitude of the periodic forcing.
Here, μ is a bias current acting on the cell, τ is the decay time, A and ω are the amplitude and (angular) frequency of the sinusoidal current acting on the cell, σ is the strength of the stochastic fluctuations, is a standard Wiener process, and denotes the right limit taken at . A spike occurs when the membrane voltage crosses a voltage threshold, , and then is instantaneously reset to the resting potential . The difference between subsequent spike times, , is called the interspike interval (ISI).
see the discussion in , which can be directly inferred from the solution in Eq. (12) below. In both the supra-threshold and supersinusoidal regimes, . The difference between the two is that in the supra-threshold regime the constant bias current alone is sufficient for spikes to occur, also in absence of noise, that is, . In the supersinusoidal regime, the sinusoidal current is necessary for spikes to occur in absence of noise, that is, and . In the critical regime, the sum of the two terms is just barely enough to guarantee deterministic spiking, that is . Finally, in the subthreshold regime, there would be no spikes without the noise, .
Example of α, β, γ parameter values for the different regimes, given
With regards to Figs. 2 and 3, it is worth noting explicitly that combinations of noise and sinusoidal forcing can cause firing patterns in which spikes are phase locked, but skip a certain number of cycles. This leads to multimodal ISI densities. There are many different dynamical mechanisms that can yield such patterns, and the particular correlations between the ISIs will depend on the underlying voltage dynamics (which, in our case, we assume to be given by Eq. (1)); in particular, it may be difficult to distinguish whether the dynamics are subthreshold or supra-threshold, since both can show similar ISI densities; see .
2.1 Basic ISI Probability Density Functions
The subscript ϕ is to stress that g, G, and depend on the value of in Eq. (3). This is the formal statement that in a sinusoidally-driven neuron, the interspike intervals are not identically distributed, and are only independent conditioned on the sinusoidal phase at an interval’s onset. Knowing these distributions would provide the likelihood function, offering estimation by the preferred method of choice, the maximum likelihood estimator. Unfortunately, explicit expressions for the ISI distribution are not available except for the special case of and ; see . Different representations of the likelihood function are available though, see , one of which we will use below.
2.2 Fokker–Planck Equation with Absorbing Boundaries
Equation (7) forms the basis of one of the methods below for estimating the structural parameters from the observed data.
This choice requires some explanation. In the limit, the distribution of in Eq. (3) without thresholding is Gaussian with mean given by Eq. (12) (below) and variance equal to . Thus, to truncate the computational domain for the thresholded process from below, we take the lowest value of the asymptotic mean, , then from this we subtract two standard deviations, and set the result to be the lower bound, . Finally, if this value for happens to be larger than −0.25, we enforce that .
Numerical considerations lead us to solve for F, instead of f, since delta functions are difficult to represent in floating point, while the initial conditions for F, the Heaviside step function, , faces no such difficulties . The Heaviside step function is defined to be equal to 0 for and to be equal to 1 for . At this point, we need to derive the PDE for the distribution F, starting from the PDE for the density, f, Eq. (5).
The right-hand side in Eq. (8) is precisely the reflecting boundary condition on f once we recall that . Therefore, .
2.3 Fortet Equation
The left-hand side is simply the probability of exceeding at time t starting at at time 0. This can also be written as the probability of hitting for the first time at time and then exceeding at time t starting at at time τ, integrated over all τ.
is the conditional cumulative distribution function of defined in Eq. (13).
3 Parameter Estimation Algorithms
The unknown parameters in Eq. (3) are α, β, and γ, while we assume Ω known. The reason why the amplitude, γ, is often unknown while the frequency, Ω, is known is that one can usually observe the sinusoidal input and thus its frequency. Further, the encoding of the input into neuronal firing patterns often involves phase locking to the sinusoidal component. However, the actual forcing amplitude at the level of the neuron is usually modified by various synaptic and other filtering processes, unless the cell receives direct sinusoidal current injection.
Our goal is to estimate the structural parameters (α, β, γ) from a sample of spike time data, . There are several algorithms for estimating the parameters for the simpler and more common case of . One such algorithm relies on the Fortet equation (see [7, 8]), which we extend to the presence of a time-varying current. A more basic approach is to directly solve the Fokker–Planck equation for the probability density of , [19–21], from which one can derive the survival distribution of , and use this to compare against the empirical survival distribution of obtained from data. An approximate maximum likelihood approach is also possible by numerical differentiation. The relation between Fokker–Planck equations and the first-passage time problem is discussed in most introductory books on stochastic analysis; see, for example, . A recent review of this approach for the simple case in neuronal modeling can be found in , wherein the first passage problem is discussed at great lengths in the context of spiking neurons. We will use this in Sect. 2.2. A more elaborate approach using the Fokker–Planck equation to approximate the hitting time distribution is given in . The techniques in  avoid the need to compute the Fokker–Planck PDE numerically, instead approximating it with analytically known solutions. This approach might offer significant computational savings, but since this would at most amount to a computational speed-up of our algorithm, we have left this unexplored for now.
The immediate problem in generalizing the aforementioned approaches to the case of is that the ’s are no longer identically distributed since the phase of the n th interval depends on , the time the previous spike occurred. The ’s are also dependent, but conditionally independent given . So the trajectories in each interval are parameterized by the value of at the time of the last spike/reset. We overcome this obstacle by splitting the ’s in groups, and approximating the ’s within groups as coming from identically distributed trajectories in a sense to be specified below. This approximation which solves the challenge of dependent and non-identically distributed ISIs is the primary contribution of this paper.
Before we can use Eq. (9) or (15), we need to deal with the fact that ϕ is not fixed, but instead each starts with a distinct . Our approach is to partition the interval into M bins, where , and represent each bin by the midpoint of the bin, . Then we approximate the N observed ’s by the closest and pretend that any observed was not produced by a trajectory of the form in Eq. (3) with , but with . Our hope is that for a judicious choice of M, we can balance the error of with having enough data points in each bin in order to obtain a useful estimate from Eq. (9) or (15).
3.2 Fokker–Planck Algorithm
The weight is included so that bins with larger sample sizes have a larger influence on the estimates.
where the derivative has to be approximated by finite differences. We can then again use binning to avoid having to compute the PDE separately for each . Our experience with the MLE approach has been that the quality of the estimates provided are similar to those obtained by minimizing Eq. (17) and that the associated computing times are on the same order. Due to this similarity and in order to keep the paper concise, we include details of the MLE estimates only in the supplementary online material.
3.3 Fortet Algorithm
We divide each inner term by , following the suggestion in . This scaling ensures that Eq. (15) divided by will vary between 0 and 1 for all parameter values thus giving sense to the measure defined by the loss function. Since we can solve in closed form for Φ, we have all we need given an observed spike train of ’s. We evaluate the sup by sampling at uniformly spaced points in and taking the maximum of the sampled values.
3.4 Initialization of the Algorithms
Its solution given an initial condition will be a Gaussian bell moving to the right with speed U and standard deviation .
for the initializer. We can form these equations separately for each bin, thus resulting in equations for the unknowns α, β, and γ. Since we have more equations than unknowns, we use least-squares estimates in a regression to pick out unique α, β, and γ estimates.
4 Method Comparison on Simulated Data
We will now use our algorithms on spike trains simulated from the four different regimes: the supra-threshold, the critical, the supersinusoidal and the subthreshold. We have used 100 sample spike trains per regime, with as well as spikes per train. In order to perform the numerical minimization of Eqs. (17) and (18), we have used an implementation of the Nelder–Mead algorithm from the SciPy library . The Nelder–Mead algorithm is a non-linear minimization routine which uses a bounding-polygon method to zero-in on the minimum and thus avoids the need to provide the gradient of the loss function. It is the standard non-gradient minimization algorithm.
Averages and empirical 95 % confidence intervals of the estimates for spikes per train
α = 1.40
β = 0.30
γ = 0.14
α = 0.10
β = 0.30
γ = 1.98
α = 0.50
β = 0.30
γ = 0.71
α = 0.40
β = 0.30
γ = 0.57
Averages and empirical 95 % confidence intervals of the estimates for spikes per train
α = 1.40
β = 0.30
γ = 0.14
α = 0.10
β = 0.30
γ = 1.98
α = 0.50
β = 0.30
γ = 0.71
α = 0.40
β = 0.30
γ = 0.57
Average times ± standard deviations in seconds for the algorithm in various regimes. Left: spikes; right: spikes
1.29 ± 0.72
0.52 ± 0.21
0.83 ± 0.28
0.18 ± 0.20
0.94 ± 0.42
0.36 ± 0.16
1.36 ± 0.46
0.43 ± 0.17
9.68 ± 4.98
1.69 ± 0.91
3.90 ± 1.05
0.21 ± 0.06
10.03 ± 2.88
1.28 ± 0.41
10.13 ± 2.24
1.06 ± 0.33
5 The Effect of Ω
So far, we have held Ω constant and equal to 1. We now investigate the effect of varying Ω on the quality of estimates. To narrow the scope, we focus on increasing Ω while keeping the parameters in the critical regime such that and . This amounts to increasing γ with Ω. We do the estimations for four values of . Similarly to the previous section, we use 100 sample spike trains per parameter set, with each spike train consisting of ISIs.
Averages and empirical 95 % confidence intervals of estimates for spikes per train in the critical regime for varying Ω across . Note that the upper subtable corresponds to the third subtable in Table 3; numbers differ slightly due to statistical fluctuations in the simulations
Ω = 1
α = 0.50
β = 0.30
γ = 0.71
Ω = 5
α = 0.50
β = 0.30
γ = 2.55
Ω = 10
α = 0.50
β = 0.30
γ = 5.02
Ω = 20
α = 0.50
β = 0.30
γ = 10.01
6 Discussion and Outlook
We have shown two methods to estimate parameters in Eq. (2) from ISI data. Our methods are based on binning the spikes into bins with representative phase shifts. We have devised a constructive procedure to automatically initialize the methods from the data.
Our computational results suggest that for low frequencies the Fortet algorithm is superior for large sample sizes, especially in the supersinusoidal regime, while the Fokker–Planck algorithm has a comparable accuracy and a lower variance for small sample sizes. Both algorithms find sensible estimates most of the time, although they seem less effective in the subthreshold regime. Their performance can be partially attributed to the ability of the initializer algorithm to supply good guesses for starting the optimization iterations.
The Fokker–Planck equation allows for approximate maximum likelihood estimation. We chose an alternative loss function, though, because it marginally appeared more robust, possibly because a numerical derivation step is avoided. This is further investigated by simulations in the supplementary online material. The simulations suggest that the distribution of the maximum likelihood estimates in the supersinusoidal regime appears bimodal, which is not the case for the alternative loss function, Eq. (17).
We have also made a preliminary exploration of the effect of Ω on the quality of the estimates. Our results show that an increase in Ω makes the parameters α and γ more difficult to estimate accurately and at high Ω, γ is underestimated, while α is over-estimated. We find that in this scenario, the Fortet algorithm does a markedly more accurate job then the Fokker–Planck algorithm.
We have assumed the time-constant τ of the leak term to be known. In most experiments that is not realistic, and it would be preferable to estimate τ alongside the other parameters. However, it is difficult to estimate . When we tried to estimate it together with the other parameters, we usually obtained results which were not accurate. The obtained estimates resulted in ISIs that very well matched the data, no worse than the ISIs obtained from the true parameters. This leads us to believe that the simultaneous estimation of τ along with α, β, γ using only ISI data suffers from identifiability problems. In , they were able to estimate τ in the simpler nonsinusoidally-driven model, but concluded that adding τ as an unknown dramatically reduced the accuracy in the estimation of the other unknown parameters. The reason is that if τ is also estimated from a single dataset alongside the other parameters, then a reasonable fit can be found to the data for various combinations of α, β, γ, and τ, but the so-obtained parameter values can be far from the true values.
Our model is relatively simple and ignores neurophysiological realism, such as the fact that the spiking threshold is likely nonconstant, with a time-dependent functional form that would involve further unknown parameters. A recent paper attempting the parameter estimation in such a model, but without sinusoidal forcing, is . Furthermore, intracellular recordings suggest that a hard threshold is a rough approximation and an exponential voltage-dependent spiking intensity is more realistic .
While our work has used a very specific form of the periodic forcing term, namely , it is clear how to apply the approach to an arbitrary periodic function. This can be done as long as one knows where in the period of oscillation a spike has occurred. If that is the case, then the binning procedure can be applied and the estimation methods proposed can be attempted.
SD is supported by the Danish Council for Independent Research|Natural Sciences. AL acknowledges support from NSERC Canada. AI is supported by a PGS-D scholarship from NSERC Canada. The work is part of the Dynamical Systems Interdisciplinary Network, University of Copenhagen.
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