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Edited by: Arjen Van Ooyen, VU University Amsterdam, Netherlands

Reviewed by: Bruce Graham, University of Stirling, UK; Szymon Leski, Nencki Institute of Experimental Biology, Poland

*Correspondence: Eoin P. Lynch, School of Mathematics, Trinity College Dublin, Dublin 2, Ireland

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Spiking neuron models can accurately predict the response of neurons to somatically injected currents if the model parameters are carefully tuned. Predicting the response of

It is difficult to find parameters for neuronal models that accurately replicate the responses seen in

Neuronal models vary in complexity from the integrate-and-fire model (Lapicque,

The various adaptive models have different strengths both in reproducing different types of spiking behavior and in ease of parameter estimation (Touboul,

Several different approaches to fitting neuron models to electrophysiological data have been tested; (Van Geit et al.,

The choice of optimization method is highly dependent on the model and the form of the experimental data, for example whether the data consists of spike trains or some other neuronal signal. If the input current to a neuron is not only known, but can be manipulated, model parameters can be estimated in an analytical fashion. For example, in Brette and Gerstner (

However, to develop a general purpose algorithm for neuron model optimization, a global heuristic optimization method is needed. This is a method which finds the global minimum, or maximum, of a function in a search space without using gradient information but instead using some carefully designed search heuristic. Examples of global heuristic search methods include stochastic search techniques such as simulated annealing and evolutionary algorithms.

In Clopath et al. (

Other fitness functions available for spike train data include the van Rossum distance (van Rossum,

The van Rossum metric, which will be applied to neuron model fitting here, is a similarity measure which is computed by convolving spike trains with a filter to produce square integrable functions and then taking the standard ^{2} distance between these functions. The measure has several useful properties; it is continuous, which gives the algorithm sensitivity to small improvements in model accuracy; it does not require binning of spikes so it retains high fidelity to the temporal structure and it satisfies all the properties of a formal metric. With this in mind, it seems likely the van Rossum metric is as useful as many of the other available measures.

In this paper a general purpose evolutionary optimization routine is presented for calculating parameters for spiking neuron models responding to time varying signals which may be an input current waveform or an auditory stimulus. The optimization method is based on the genetic algorithm with real-value gene representations. The algorithm uses the van Rossum spike train metric as the fitness function.

Initially the algorithm is tested by fitting spiking neuron models to synthetic target data generated by other models with parameters known to lie in the search space. The algorithm is then applied to fitting models to experimental spike train data. Several two variable spiking neuron models are fit to

The algorithm is then extended to optimize an auditory neuron model, consisting of a cascade of a receptive field and a spiking neuron, by a tandem evolution approach. The model is fit to data consisting of song stimuli and extracellular spike train recordings from Zebra Finch. The full auditory model is compared to linear rate models in predicting the activity of the Zebra Finch auditory neurons responding to conspecific songs using two measures: the average coincidence factors and the van Rossum distances across trials.

In this section we review the standard spiking neuron models and introduce a new spiking model which we found to be effective when applied to experimental data. Spiking neuron models, in contrast to more biophysically grounded models such as the Hodgkin-Huxley or Morris-Lecar neurons, are phenomenological models that reproduce neuron-like behavior with relatively few parameters. They exhibit neuron-like behavior in response to excitation but usually are not rooted in a careful balance of terms representing voltage gated ion channels acting on varying timescales, as in the Hodgkin-Huxley model. Rather, a sharp, discontinuous reset process at a spike threshold is used to model the down-sweep of the action potential.

This work investigates the fitting of models of this type with two variables. These models are expressed as either one or a pair of coupled ordinary differential equations with the general form as below. The first equation describes the time course of the membrane voltage

Here,

Since these models have no mechanisms to restore the voltage to resting after a spike, a sharp reset condition must be imposed at a cut-off voltage, _{c}, so when _{c}

The voltage is reset to a reset value, _{r}, and the adaptation variable is incremented by α.

In a real neuron, the membrane voltage rises sharply to a value greater than zero after a threshold is reached due to the opening of voltage-gated sodium ion channels; this effect is reproduced somewhat in exponential and quadratic integrate-and-fire models but not in the simple leaky integrate-and-fire model where there is no non-linearity around the voltage threshold. In other words, the exponential and quadratic models describe some part of the upswing of the spike but still require the manual insertion of part of the upswing and the down swing, the leaky integrate and fire model does not model any of the spike and do not account for the dynamics near threshold.

The various models in the literature are obtained by an appropriate choice of function

where τ_{m} is the membrane time constant. The aIF model is defined by

where τ_{w} is the adaptation time constant.

A separate but related type of model, involving an adaptation of the spiking cutoff voltage replaces the second equation in the previous pair, Equation (2), with a similar equation for _{c},

and instead incrementing _{c} during the reset process by α.

The aEIF model is defined by

_{L} is the effective resting potential of the neuron and Δ_{T} is a slope factor for the exponential term. _{T} is the threshold voltage. When

The adaptive quadratic integrate-and-fire (aQIF) or Izhikevich model (Izhikevich,

and a reset condition as before although Izhikevich refers to the reset voltage as _{R} and the increment in the adaptation variable as

Possible extensions of these two variable models are inclusions of multiple time scales of adaptation in either the threshold or the adaptation current by the inclusion of additional equations of the form of Equations (2) or (10). Promising results have been achieved with an approach like this. Kobayashi et al. (

Overall, it seems that adaptation is both crucial to model performance and occurs on multiple time scales. In Kobayashi et al. (_{i} is used to calculate an effective predictive score, Γ_{A} = Γ/Γ_{i}. The Hodgkin-Huxley model achieved a predictive score of Γ_{A} = 0.51 ± 0.26. The leaky integrate-and-fire model and spike response model with a single scale adaptive thresholds yielded Γ_{A} = 0.66 ± 0.26, and Γ_{A} = 0.70 ± 0.26, respectively. The multiple adaptive timescales model with three scales of adaptation achieved the best predictive score of Γ_{A} = 0.89 ± 0.21. In otherwords, the effectiveness of the model increases as the amount of adaptation is increased and the inclusion of adaptation is more useful at improving accuracy than the biophysical replication of short timescale channel dynamics of the Hodgkin-Huxley equation.

Here we propose an extension of the aEIF model which includes a second level of adaptation in the spike threshold parameter. The model has similar sub-threshold dynamics to the original aEIF model, however the extra dynamical equation allows for the threshold to increase after spiking. We call this multiply adaptive exponential integrate-and-fire model the a^{2}EIF model. The dynamical equations of the model are

The parameters τ_{t} and β are the time constant of threshold adaptation and spike-threshold increment respectively. The reset occurs when the voltage reaches _{c} and has an effect on each of the three dynamic variables;

where _{t} parameter defines the membrane voltage at which the exponential term becomes positive and initiation of a spike is almost certain.

In this section the other part of modeling the

The responses of auditory neurons to auditory stimuli are commonly characterized using spectro-temporal receptive field (STRF) models (Aertsen and Johannesma,

In the linear case, the receptive field is estimated by formulating a least squares problem between the estimated firing rate,

Several methods have been applied to the solution of Equation (20), with normalized reverse correlation being commonly used (Theunissen et al.,

The STRF calculation used here is based on the pseudo-inverse technique commonly used elsewhere (Theunissen et al.,

where

where _{1}, …, _{M × N}]^{T} is the vector of STRF coefficients. Minimizing

where (^{T}_{sr} is the cross correlation vector between the stimulus and the response. The inversion of the stimulus auto-correlation matrix requires careful consideration.

The singular value decomposition of the auto-correlation matrix (^{T}^{T}^{−1} is computed by first taking the singular value decomposition of ^{T}^{T}

where Σ is a diagonal matrix of the singular values and ^{T}

A regularization strategy is applied to the singular value matrix, Σ, during inversion to form the regularized pseudo-inverse. It requires selection of a single hyper-parameter λ which specifies the tolerance of the regularization and is optimized using a cross-validation procedure. The number of stimulus dimensions to preserve,

If σ_{i} are the singular values in Σ, then after this regularization procedure the non-zero elements of the inverse of Σ, that is Σ^{+}, are given by

Receptive field models estimate firing rates and not spike trains. However, the firing rate can be used to generate spike trains using some spike generation mechanism. For example, an inhomogeneous Poisson process can be used to generate spike trains in a probabilistic manner from the rate. Here, a two stage model is used in which a receptive field model is adapted to produce an estimate of the input signal to a two-dimensional spiking neuron model. This is a deterministic model of spike generation for auditory neurons

We shall attempt to fit this model to ^{2}EIF model will not be used in the two stage auditory model because it presents a more challenging optimization problem, as will be shown later, and its parameters have not been well studied. Instead, the standard aEIF neuron will be used.

In this paper a genetic algorithm is used; it is customized to parameter discovery for the two stage spiking model. The genetic algorithm is a heuristic optimization method inspired by Darwinian evolution, first introduced in Holland (

More precisely, the genetic algorithm is used to find a parameter set ^{n} that minimizes a fitness function ^{n}. It is an iterative, population-based method in which a population set _{0} = {_{1}, …, _{N}} of candidate parameter sets, _{i} ∈ _{1}, …, _{i} are iteratively generated from the previous generation using a number of pre-defined rules to combine and randomly modify parameter sets and form new parameter sets.

The fitnesses of the parameter sets in each generation are calculated and ordered in ascending order. To create a new generation set, _{1}, first, the fittest _{0} are selected and copied exactly to form _{1}. This procedure is referred to as

Next, breeding is used to replace the other _{1} are randomly selected with some mutation probability, typically ≈5%, to undergo mutation.

The algorithm developed here uses a non-uniform mutation operator inspired by those commonly presented in, for example, Michalewicz (_{m}. The value of ϵ_{m} decreases with successive iterations. This causes the algorithm to search the parameter space widely in early iterations and to more finely tune its search in later generations. 0.2 was determined to be a roughly optimal initial value for ϵ_{m} through trial and error as, on average, the best convergence was observed with this value.

The differential equations governing spiking neuron models are invariant under a number of scaling transformations. Indeed many of the models can be rescaled and their number of parameters reduced. For example, Touboul and Brette (

There is also an ambiguity in the overall scale of ^{1} norm equal to one. The correct scaling factor is then regarded as one of the parameters describing the neuron model and is optimized along with the other neuronal parameters.

The parameters of each model that was presented in Section 2.1 and 2.2. are summarized in Table

atIF | τ_{m}, τ_{t}, α, |
_{L}, _{c0} |
5 | 20 bytes | |

aIF | τ_{m}, τ_{w}, α, |
_{L}, _{c} |
4 | 16 bytes | |

aEIF | τ_{m}, τ_{w}, _{L}, Δ_{T}, _{T}, _{r}, |
_{c} |
9 | 36 bytes | |

^{2}EIF |
τ_{m}, τ_{w}, τ_{t}, _{L} Δ_{T}, _{r}, _{t0}, |
_{c} |
11 | 44 bytes | |

aQIF | _{c} |
4 | 16 bytes | ||

STRF | 20× 40 matrix |
800 | 3200 bytes |

Here the van Rossum metric (van Rossum,

is mapped to a real function,

Here, an exponential kernel is used:

where τ is a timescale which determines the relative sensitivity of the metric to fine temporal features in the spike trains. The van Rossum distance between two spike trains ^{2} metric between their respective functions, that is

An efficient algorithm for computing the metric, presented in Houghton and Kreuz (

Often, some sort of metric clustering based optimization routine is used to pick the best timescale τ for a data set (Victor and Purpura,

It is convenient to introduce objects that play the role of “average” spike trains. Following Julienne and Houghton (

In addition to the van Rossum metric, we use the the coincidence factor as benchmark test for quantifying the performance of neuron models since it is widely used for this purpose (Jolivet et al.,

where _{c} is the number of coincident pairs between the two spike trains, _{e} and _{m} are the number of spikes in the experimental and model spike trains respectively. δ is the coincidence window which defines a coincidence; if the absolute time difference between two spikes is less than δ then they are coincident. _{i}, is the average inter-trial coincidence factor. It is a measure of the underlying unreliability of the spike trains and can be used to rescale model coincidence factors into an effective performance factor. If a model achieves a performance factor of Γ/Γ_{i} ≥ 1 in predictions on a validation set then it can safely be concluded that the model is making predictions to within the variability of experimental data set.

A number of numerical experiments were designed and performed to test the effectiveness of our algorithm at parameter estimation and neural response prediction in different situations. The details of these experiments are described in this section.

To test the effectiveness of the algorithm in isolation, synthetic target data generated by a spiking neuron model target was used. Here it was expected that a model with a high coincidence factor with the target data could be found by the algorithm since it was known in advance that a model with the same dynamics exists.

Twenty experimental runs were performed as follows. Using the aEIF model, a sample parameter set was used to simulate 4 s of spike train data in response to a random input current signal. This synthetic data set, consisting of the spike train and current signal, was then used as the target for the genetic algorithm. For this purpose, the data set was divided in two with the first 2 s being for training and the final 2 s for validation. The genetic algorithm was initialized with a population of 240 neuron models each with a parameter set drawn randomly from its feasible region. The target parameters are indicated in Table

_{m} |
_{w} |
_{T} |
_{R} = _{L} |
_{T} |
|||
---|---|---|---|---|---|---|---|

Range | [3, 17] ms | [36, 204] ms | [0.0003, 0.0017] | [−70, −20] mV | [−120, −50] mV | [0.3, 1.7] mV | [0.5, 3] mV |

Target | 10 ms | 144 ms | 0.001 | −50 mV | −70 mV | 1 mV | 2 mV |

Discovered | 11 ± 2.6 ms | 145.5 ± 4 ms | 0.0011 ± 0.0004 | −50 ± 7 mV | −70 ± 1 mV | 1± 0.1 mV | 1.75 ± 0.4 mV |

We chose a population size of 240 as it is the number of cores on the NVidia CUDA GPU we ran the code on. The algorithm was set to run for 1000 iterations to allow full convergence and the evolution of the population best van Rossum distance and coincidence factor when compared against the validation data set was observed.

It was also investigated here whether using a variable timescale in the van Rossum distance fitness function results in a more effective optimization search than a fixed timescale. Three test case optimization runs were performed; in the first the algorithm was set to use a timescale which starts at a value half the length of the target data set and is reduced with subsequent iterations to a value equal to the mean inter-spike interval. The decrement in τ is logarithmic with the base being calculated using the maximum number of genetic algorithm iterations,

The model fitting procedure was tested on the publicly available data from challenge A of the INCF quantitative single neuron modeling competition, 2009 (Gerstner and Naud,

The 20.5 s segment of current injection data set was divided up as follows; the first 10.5 s was used for model fitting and the the last 10 s for validation. The spike times were extracted from the voltage trace by using interpolation to find the times in the trace at which the voltage crossed a threshold value.

The algorithm was run for each of five neuron models to find the best parameters in each of the aIF, atIF, aEIF, a^{2}EIF, and Izhikevich models. The algorithm was set to run for 800 iterations on each trial. The best average inter-trial coincidence factor obtained from each run and the corresponding parameters were recorded. We also studied the MAT model using the parameters described in Kobayashi et al. (

For testing the auditory model, we used the publicly available Zebra Finch data set, “aa-2,” available on the collaborative research in computational neuroscience (CRCNS) website (Gill et al.,

Data consisting of 110 of the 445 electrode data sets was selected from the full set for testing of the full model optimization routine as follows; an ensemble average intrinsic reliability, 〈Γ_{i}〉, was first estimated for each of the 445 cells in the data set by computing the intrinsic reliability between trials for each stimulus with δ = 2 ms and then averaging across stimuli. The cells with 〈Γ_{i}〉 greater than 0.1 were then selected for testing. A coincidence factor of 0 would correspond to random chance so 0.1 was deemed a sensible limit to eliminate highly noisy data.

The spectro-temporal receptive field of each cell in the cohort was computed using the normalized reverse correlation method presented earlier using the STRFlab Matlab toolbox (David et al.,

Finally, the full model optimization method was applied to the 110 data sets from the experimental data set as follows. First, each data set was split in half to form training and validation sets. For example, in the sets with 20 songs, 10 were used for validation and 10 for training. The algorithm was run for a maximum of 600 iterations on each cell, using the average van Rossum function obtained by summing all the spike trains as the target. It was observed that the algorithm generally stopped converging after a few 100 iterations. This is how a limit of 600 iteration was decided upon; no significant improvement was likely beyond this.

The performance of the full model spike predictions were benchmarked against the STRF rate models realized as a set of spike trains by generating spikes from the rate using an inhomogeneous Poisson process (Lewis and Shedler,

The expected number of spikes generated by the inhomogeneous Poisson process, 〈_{m}〉, is just the integral of the rate over the time _{coinc}〉 was estimated using a Monte Carlo like method in which the number of coincidences was evaluated for 100 simulated Poisson spike trains with rate

Expected van Rossum distances between the inhomogeneous Poisson neurons and the target spike trains were calculated. This can be accomplished by first filtering the Poisson neuron firing rate with the exponential filter of the van Rossum distance. The resulting function is equivalent to the expected function obtained by filtering and averaging very large ensemble of Poisson spike train realizations of the rate. Thus, the expected Van Rossum distance between a Poisson process with rate

An evolutionary algorithm for fitting spiking neuron models to time varying signals has been presented. The system used a van Rossum metric between spike trains as a fitness function. Initially, the algorithm was applied to synthetic data to study its convergence properties. The algorithm was then applied to modeling

The purpose of testing the algorithm on artificial target data was to show that the algorithm was capable of finding, near enough, a set of parameters which are known to exist somewhere in the search space. Figure

The target parameters and the values obtained by the search algorithm are given in Table

The effect of varying the van Rossum distance timescale on model convergence was investigated during the runs on artificial target data. In Figure

Using a varying timescale consistently improved convergence behavior over using a constant timescale. A short τ resulted in a rapid initial climb which leveled off relatively early. A long τ resulted in a slower rise in coincidence factor. The varying time scale allows the correct firing rate to be quickly found in the early stages of the search by using a wide timescale, leading to sharp improvements early on. Later, with a narrower timescale, the van Rossum distance switches to functioning more like a coincidence detector and fine tunes the model. The effect of a varying timescale on the resulting spike train functions is illustrated in Figure

The performance results of the model fitting studies on the

_{i} |
^{o} Parameters |
||
---|---|---|---|

aIF | 0.63 ± 0.04 | 4 | 1.7 |

atIF | 0.64 ± 0.06 | 5 | 1.8 |

aEIF | 0.74 ± 0.06 | 9 | 8.3 |

a^{2}EIF |
0.78 ± 0.03 | 11 | 8.4 |

aEIF approximation | 0.74 ± 0.06 | 9 | 5.6 |

a^{2}EIF approximation |
0.77 ± 0.03 | 11 | 5.7 |

MAT^{*} |
0.59 ± 0.02 | 5 | 1.1 |

Izhikevich | 0.38 ± 0.03 | 5 | 2.1 |

^{2}EIF models were obtained using an approximation of the exponential term as described in Schraudolph (

The a^{2}EIF gave the highest predictive accuracy among the models studied. The computational cost of the model is only marginally higher than the aEIF model as the extra equation is linear. The aEIF model does however converge to 90% of the maximum achieved in considerably less iterations of the genetic algorithm than the a^{2}EIF model. This is illustrated in Figure

In Table

This approximation was tested. The performance and computational cost figures obtained with this method are shown in the table in brackets next to the figures for the aEIF and a^{2}EIF neurons obtained using calls to the standard exponential function in the C math library. A speed increase of more than 30% is achieved with negligible change in predictive accuracy according to the coincidence factor calculated at a resolution of 2 ms. However, the computational cost is still more than five times that of a leaky integrate-and-fire neuron.

The MAT model was less computationally costly to simulate than the other models we studied as it is amenable to an analytical solution. We used the code published by Kobayashi et al. (

The values of best coincidence factor and the best van Rossum distance obtained from the STRF-aEIF model on the validation data set are plotted in Figure

^{+}refer to the individual data points; each ^{+}corresponds to an individual cell in the cohort of cells studied.

For many of the cells, the algorithm found solutions close to these theoretical values of the reliability of the data, particularly so for van Rossum distances. These limits are not hard upper limits however and models can exceed them in certain circumstances. For example, if a model spike train is found which lies somewhere in the center of a space whose edges are defined by the set experimental spike trains, then that spike train can have a lower average distance from all the experimental trains than the average inter-trial distance between the experimental trains. Such a scenario is likely to occur when the experimental recordings are particularly noisy and variable.

The predictive accuracy of the cascade STRF-aEIF model vs. normalized reverse correlation STRFs realized as spiking neurons using an inhomogeneous Poisson process is illustrated in Figure

^{+}refer to the individual data points; each ^{+}corresponds to an individual cell in the cohort of cells studied.

The aEIF neuron model parameters obtained by fitting the STRF-aEIF model are collected and the distribution of some of these parameters are plotted in histograms in Figure _{t}, determines how big of an impact the exponential term has on the dynamics of the model. Many of the estimated values of Δ_{t} are quite close to zero and less than the sample value quoted by Brette and Gerstner (

_{T}/_{m}/τ_{w} and _{T}, in the aEIF model.

The runtime of the algorithm was on the

An optimization tool for generating models of spiking neurons from both

The van Rossum distance was used as a fitness function. A continuous objective function is useful in a continuous parameter space since incremental steps toward convergence lead to incremental improvements in the value of the function. We found that the choice of timescale in the van Rossum distance made it an adaptable fitness function. A large timescale quickly narrows the search to models with the correct firing rate while a shorter timescale allows fine tuning of the model parameters.

Several other authors have studied optimization methods for spiking neurons. Gerken et al. (

Our results on the ^{2}EIF model presented here yielded the greatest predictive accuracy but was the most costly to simulate and second most costly to fit. As noted elsewhere (Jolivet et al.,

When we applied our algorithm to fit a STRF-neuron cascade model to

The optimized aEIF neuron model parameters do not vary very much across all the cells studied: Figure

An example raster plot showing a model response and corresponding experimental spike train responses from a typical cell from the

A drawback of our method for fitting the

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We are grateful to Frederic Theunissen, Patrick Gill, Amin Noopur, Junli Zhang, Sarah M. N. Woolley, and Thane Fremouw for making their Zebra Finch data available through the Collaborative Research in Computational Neuroscience website (Theunissen et al.,

The Supplementary Material for this article can be found online at: