Synapses are key elements in the information transmission in the nervous system. Among the different approaches to study them, the use of computational simulations is identified as the most promising technique. Simulations, however, do not provide generalized models of the underlying biochemical phenomena, but a set of observations, or timeseries curves, displaying the behavior of the synapse in the scenario represented. Finding a general model of these curves, like a set of mathematical equations, could be an achievement in the study of synaptic behavior. In this paper, we propose an exploratory analysis in which selected curve models are proposed, and stateoftheart metaheuristics are used and compared to fit the free coefficients of these curves to the data obtained from simulations. Experimental results demonstrate that several models can fit these data, though a deeper analysis from a biological perspective reveals that some are better suited for this purpose, as they represent more accurately the biological process. Based on the results of this analysis, we propose a set of mathematical equations and a methodology, adequate for modeling several aspects of biochemical synaptic behavior.
Most information in the mammalian nervous system flows through chemical synapses. These are complex structures comprising a presynaptic element (an axon terminal) and a postsynaptic element (a dendritic spine, a dendritic shaft, an axon, or a soma) separated by a narrow gap known as the synaptic cleft (see Figure
Chemical synapses. (a) Electron microphotograph of the cerebral cortex where three axon terminals (asterisks) establish synapses with three dendritic elements showing clearly visible postsynaptic densities (arrows). (b) Simplified model of a chemical synapse, highlighting its most significant parts.
Electron microscopy image
Schematic representation
Multiple factors influence the diffusion of neurotransmitter molecules and their interaction with specific receptors [
Simulation approaches in neuroscience have considered different models, scales, and techniques, according to the phenomenon being studied. Biochemical processes, such as neurotransmitter diffusion, require Monte Carlo particlebased simulators like MCell [
In the case of chemical synapses, many aspects of their behavior during neurotransmitter release and diffusion can be simulated. Simulations, however, do not provide generalized models of these series, showing only the specific values observed each time and, in this case, these are subject to the stochastic nature of the Monte Carlo simulations used to produce them. Having general, empirical models of these time series, such as a set of mathematical equations, could be an important achievement in the study of chemical synaptic behavior. Finding these equations, however, is not a trivial task.
Having the raw simulation data at hand, the first step in finding these equations is to identify the type of mathematical expression we are looking for. A reasonable approach to achieve this is to study the underlying physicochemical processes that take place during synapse operation (e.g., molecular diffusion and receptor kinetics) and try to analytically infer the synaptic equations from them. This is, however, sometimes not possible (as is explained in detail in Section
This problem is, in its general form, an optimization task, in which different search strategies in the field of metaheuristics have been successfully applied in the past. This type of search techniques, with Evolutionary Algorithms (EAs) as one of the most relevant exponents, have shown in the last decades their ability to deal with optimization problems of many different domains, ranging from completely theoretical mathematical functions (benchmarks) [
We analyzed simulations based on simplified models of excitatory synapses where AMPA receptors were present and the neurotransmitter involved was glutamate (The AMPA receptor is a ionotropic receptor commonly found in excitatory synapses using glutamate as neurotransmitter. It is named from its specific agonist alphaamino3hydroxy5methyl4isoxazolepropionic acid). We based our study in a corpus of synaptic simulations very similar to the one presented in [
The average variation of neurotransmitter concentration in the synaptic cleft over time, after a single vesicle release.
The average variation of the total number of synaptic receptors in the open state over time, again after a single neurotransmitter vesicle release (Molecular kinetics of synaptic receptors follow Markovchainlike models, with several possible states and transition probabilities depending on environmental factors and chemical reaction rate constants. The most relevant state in the AMPA receptor is the open state. It is directly related with the electrical synaptic response [
These two aspects were plotted as timeseries curves for each of the 500 synaptic configurations, resulting in a total of 1,000 curves (500
Examples of synaptic curves after a neurotransmitter vesicle release. (a) Variation of neurotransmitter (in this case
Open
In the case of the
We integer both sides:
We now apply
Since
The value of
From the second item, we deduce that the
The coefficient
All open
As we have already established, neurotransmitter concentration cannot be predicted analytically (Fick’s second law cannot be solved analytically for 2 or more dimensions); therefore open
Therefore, we adopted the exploratory analysis approach, considering a set of selected general mathematical models and testing them against the simulation data. We based this set on a previous analysis presented in [
4by4 degree polynomial rational function (9 coefficients):
8term Fourier series (18 coefficients):
8term Gauss series (25 coefficients):
2term exponential function (4 coefficients):
9th degree polynomial (10 coefficients):
To decide which of these curves best serves our purpose, we need to test how well each of them can model the experimental data we observed during simulations. We need, therefore, to fit every curve to the data and see how they perform.
This section reviews the considered algorithms for the previously described curves fitting problem, including the reference baseline algorithm (Nonlinear Least Squares) used in previous studies [
The Nonlinear Least Squares (NLLS) algorithm [
To assess the accuracy of our curvefitting solutions, we based our study on the commonly used coefficient of determination (
And
The coefficient of determination, however, does not take into account regression model complexity or number of observations. This is particularly important in our case, since we want to compare the performance of very different curve models, for example, the 2term exponential function, which has 4 coefficients, against the 8term Gauss series, which has 25 coefficients. To incorporate these aspects, we adopted the adjusted coefficient of determination (
All the experimentation reported in this paper has followed a fractional design based on orthogonal matrices according to the Taguchi method [
This method allows the execution of a limited number of configurations and still reports significant information on the best combination of parameter values. In particular, a maximum of 27 different configurations were tested for each algorithm on the whole set of models and a subset of the curves (the exact number of configurations will depend on the number of parameters to be tuned). In Section
To validate the results obtained and the conclusions derived from them, it is very important to use the appropriate statistical tests for this purpose. In this paper we conducted a thorough statistical validation that is discussed in this section.
First, the nonparametric Friedman’s test [
Apart from the
Overall ranking according to the Friedman test: we computed the relative ranking of each algorithm according to its mean performance for each function and reported the average ranking computed through all the functions. Given the following mean performance in a benchmark of three functions for algorithms
nWins: this is the number of other algorithms for which each algorithm is statistically better minus the number of algorithms for which each algorithm is statistically worse according to the Wilcoxon Signed Rank Test in a pairwise comparison [
This meticulous validation procedure allowed us to provide solid conclusions based on the results obtained in the experimentation.
We have conducted a thorough experimentation to elucidate if the selected metaheuristics can be successfully used in the curvefitting problem. For this purpose, we have taken into account both
In the first place, a parameter tuning of the seven selected metaheuristics has been carried out, whose results are presented in Section
As described in Section
Parameter tuning of the selected metaheuristics.
Parameter values of GA  

popSize  100, 200, 
pcx  0.1, 0.5, 
pm  0.01, 


Parameter values of DE  


popSize  25, 50, 

0.1, 
CR  0.1, 0.5, 


Parameter values of SelfAdaptive DE  


popSize  25, 50, 






Parameter values of GODE  


popSize 


0.1, 
CR  0.1, 
godeProb  0.2, 


Parameter values of Solis and Wets  


maxSuccess 

maxFailed  1, 3, 
adjustSuccess  2, 4, 
adjustFailed  0.25, 0.5, 
delta 



Parameter values of MTSLS1  


(initial) SR  50% of the search space 
(min) SR 

adjustFailed  2, 
adjustMin  2.5, 5, 
moveLeft 

moveRight  0.25, 


Parameter values of CMAES  



0.01, 0.1, 
Active CMA 

CMAES variant  CMAES, 
Once the best configuration for each algorithm has been determined, we ran each of them on both synaptic curves problems, for all the models and curves and conducting
As we stated before, in the 500
Mean fitness, average ranking, and number of wins (nWins) in pairwise comparisons with the other algorithms on the glutamate concentration problem.
Mean fitness  Ranking  nWins  

SelfAdaptive DE 

4.16  3 
DE 

4.16  3 
GA 

4.16  3 
Solis and Wets 

4.16  3 
GODE 

4.16  3 
IPOPCMAES 

4.16  3 
MTSLS1 

4.23  −4 
NLLS constrained 

7.91  −7 
NLLS unconstrained 

7.91  −7 
Due to the nonexistent differences in the performance of most of the algorithms, the statistical validation did not reveal any significant differences, except for the two versions of the baseline algorithm. This first result is a good start point that suggests that more advanced metaheuristics can deal with these problems more effectively than the NLLS algorithm used in previous studies [
To provide more insight on the results obtained for the
The results for the 4by4 degree polynomial rational function are reported in Table
For the 8term Fourier series results are similar in terms of the dominant algorithms (see Table
The results for this model break the trend started in the previous two cases that follows in the remaining two functions. Table
In the case of the 2term exponential function, both the SelfAdaptive DE and the DE algorithms obtain very good results, the former being the best algorithm in this problem (with a precision of more than
The last function used in the experimentation, the 9th degree polynomial, is an interesting case, as we can classify studied algorithms into two groups: those that converge to (what seems) a strong attractor and those that are unable to find any good solution. In the first group we have the three aforementioned algorithms that usually exhibit the best performance: IPOPCMAES, SelfAdaptive DE, and DE, plus both NLLS configurations. Consequently the second group is made up of the GA, the GODE algorithm, and the two local searches. The algorithms in the first group normally converge, with some unsuccessful runs in the case of NLLS constrained and IPOPCMAES, to the same attractor, and thus their average precision and ranking is very similar (Table
In this section we offer an overall comparison of the algorithms for the five
To understand the biological implications of this study we have to consider each of the two curvefitting problems independently (
In the case of the
Raw
SelfAdaptive DE versus  Friedman 
Wilcoxon 

DE 


GA 


Solis and Wets 


GODE 


IPOPCMAES 


MTSLS1 


NLLS constrained 


NLLS unconstrained 


Corrected
SelfAdaptive DE versus  Friedman 
Wilcoxon 

DE 


GA 


Solis and Wets 


GODE 


IPOPCMAES 


MTSLS1 


NLLS constrained 


NLLS unconstrained 


Mean fitness, average ranking, and number of wins (nWins) in pairwise comparisons with the other algorithms on the 4by4 degree polynomial rational function.
Mean fitness  Ranking  nWins  

IPOPCMAES 

1.86  8 
SelfAdaptive DE 

2.08  5 
DE 

2.09  5 
GA 

4.86  2 
MTSLS1 

6.04  0 
GODE 

6.04  −2 
NLLS constrained 

7.04  −4 
Solis and Wets 

7.17  −6 
NLLS unconstrained 

7.84  −8 
Raw
IPOPCMAES versus  Friedman 
Wilcoxon 

SelfAdaptive DE 


DE 


GA 


MTSLS1 


GODE 


NLLS constrained 


Solis and Wets 


NLLS unconstrained 


Corrected
IPOPCMAES versus  Friedman 
Wilcoxon 

SelfAdaptive DE 


DE 


GA 


MTSLS1 


GODE 


NLLS constrained 


Solis and Wets 


NLLS unconstrained 


Mean fitness, average ranking, and number of wins (nWins) in pairwise comparisons with the other algorithms on the 8term Fourier series.
Mean fitness  Ranking  nWins  

SelfAdaptive DE 

1.26  8 
IPOPCMAES 

1.91  6 
GODE 

3.46  4 
DE 

4.34  2 
GA 

4.79  0 
NLLS unconstrained 

5.46  −2 
MTSLS1 

7.29  −4 
NLLS constrained 

7.49  −6 
Solis and Wets 

9.00  −8 
Raw
SelfAdaptive DE versus  Friedman 
Wilcoxon 

IPOPCMAES 


GODE 


DE 


GA 


NLLS unconstrained 


MTSLS1 


NLLS constrained 


Solis and Wets 


Corrected
SelfAdaptive DE versus  Friedman 
Wilcoxon 

IPOPCMAES 


GODE 


DE 


GA 


NLLS unconstrained 


MTSLS1 


NLLS constrained 


Solis and Wets 


Mean fitness, average ranking, and number of wins (nWins) in pairwise comparisons with the other algorithms on the 8term Gauss series.
Mean fitness  Ranking  nWins  

IPOPCMAES 

1.00  8 
NLLS unconstrained 

2.23  6 
NLLS constrained 

2.82  4 
GA 

3.95  2 
GODE 

5.00  0 
SelfAdaptive DE 

6.79  −3 
MTSLS1 

6.85  −3 
DE 

7.35  −6 
Solis and Wets 

9.00  −8 
Raw
IPOPCMAES versus  Friedman 
Wilcoxon 

NLLS unconstrained 


NLLS constrained 


GA 


GODE 


SelfAdaptive DE 


MTSLS1 


DE 


Solis and Wets 


Corrected
IPOPCMAES versus  Friedman 
Wilcoxon 

NLLS unconstrained 


NLLS constrained 


GA 


GODE 


SelfAdaptive DE 


MTSLS1 


DE 


Solis and Wets 


Mean fitness, average ranking, and number of wins (nWins) in pairwise comparisons with the other algorithms on the 2term exponential function.
Mean fitness  Ranking  nWins  

SelfAdaptive DE 

1.69  8 
DE 

1.86  6 
IPOPCMAES 

2.46  4 
Solis and Wets 

4.84  2 
MTSLS1 

5.43  0 
NLLS unconstrained 

5.73  −2 
NLLS constrained 

7.25  −4 
GODE 

7.68  −6 
GA 

8.06  −8 
Raw
SelfAdaptive DE versus  Friedman 
Wilcoxon 

DE 


IPOPCMAES 


Solis and Wets 


MTSLS1 


NLLS unconstrained 


NLLS constrained 


GODE 


GA 


Corrected
SelfAdaptive DE versus  Friedman 
Wilcoxon 

DE 


IPOPCMAES 


Solis and Wets 


MTSLS1 


NLLS unconstrained 


NLLS constrained 


GODE 


GA 


Mean fitness, average ranking, and number of wins (nWins) in pairwise comparisons with the other algorithms on the 9th degree polynomial.
Mean fitness  Ranking  nWins  

NLLS unconstrained 

2.44  8 
SelfAdaptive DE 

2.50  5 
DE 

2.50  5 
NLLS constrained 

2.72  2 
IPOPCMAES 

4.95  0 
GA 

7.48  −5 
Solis and Wets 

7.48  −5 
MTSLS1 

7.48  −5 
GODE 

7.48  −5 
Raw
NLLS unconstrained versus  Friedman 
Wilcoxon 

SelfAdaptive DE 


DE 


NLLS constrained 


IPOPCMAES 


GA 


Solis and Wets 


MTSLS1 


GODE 


Corrected
NLLS unconstrained versus  Friedman 
Wilcoxon 

SelfAdaptive DE 


DE 


NLLS constrained 


IPOPCMAES 


GA 


Solis and Wets 


MTSLS1 


GODE 


Mean fitness, average ranking, and number of wins (nWins) in pairwise comparisons with the other algorithms for all the models.
Mean fitness  Ranking  nWins  

IPOPCMAES 

2.44  8 
SelfAdaptive DE 

2.86  6 
DE 

3.62  4 
NLLS unconstrained 

4.74  2 
NLLS constrained 

5.46  0 
GA 

5.83  −2 
GODE 

5.93  −4 
MTSLS1 

6.61  −6 
Solis and Wets 

7.50  −8 
Raw
IPOPCMAES versus  Friedman 
Wilcoxon 

SelfAdaptive DE 


DE 


NLLS unconstrained 


NLLS constrained 


GA 


GODE 


MTSLS1 


Solis and Wets 


Corrected
IPOPCMAES versus  Friedman 
Wilcoxon 

SelfAdaptive DE 


DE 


NLLS unconstrained 


NLLS constrained 


GA 


GODE 


MTSLS1 


Solis and Wets 


Example of curvefit solution for the
It is important to remember that this equation is the solution for one particular synaptic configuration. Each one of the total 500 configurations produced a different
The case of the open
Average ranking and number of wins (nWins) in pairwise comparisons for the best algorithm for each AMPA model.
Mean fitness  Ranking  nWins  

8term Gauss series 

1.63  4 
4by4 degree polynomial rational function 

1.67  2 
2term exponential function 

2.73  0 
8term Fourier series 

3.99  −2 
9th degree polynomial 

4.97  −4 
From a strictly numerical point of view, the 8term Gauss series fitted with the IPOPCMAES algorithm produces the best results, so it is the first logical choice. In addition, we consider the nature of the biological process modeled (
To continue our analysis, we performed an exploratory graphical study of the curve solutions provided by the different optimization algorithms. Figure
The 8term Gauss, 4by4 degree polynomial rational, and 2term exponential functions are clearly the best candidates for modeling the
The curvefitting calculated for the 8term Gauss series presents an abnormal perturbation near the curve peak (see the detail in Figure
The 9th degree polynomial produces low quality solutions, clearly surpassed by the first three equations.
The 8term Fourier series seems to be overfitted to the training data. The numerical results are apparently good, which indicates that the curve points tested when calculating the fitness value are very close to the simulation ones (Figure
Examples of curvefit solutions for the
8term Gauss series (
4by4 deg. polynomial rational (
2term exp. function (
9th degree polynomial (
8term Fourier series (
8term Fourier series (tested points only)
Taking all this into consideration, we can try to select the appropriate equation for the
In the example shown in Figure
As in the case of the
In this paper we have compared several metaheuristics in the problem of fitting curves to match the experimental data coming from a synapse simulation process. Each one of the 500 synaptic configurations used represents a different synapse and produces two separate problem curves that need to be treated independently (
From a biological point of view, we have observed how several curve models are able to very accurately fit a set of timeseries curves obtained from the simulation of synaptic processes. We have studied two main aspects of synaptic behavior: (i) changes in neurotransmitter concentration within the synaptic cleft after release and (ii) synaptic receptor activation. We have proposed an analytically derived curve model for the former and a set of possible curve equations for the latter. In the first scenario (changes in neurotransmitter concentration) the optimization results validate our proposed curve model. In the second one (synaptic receptor activation) several possible curve models attained accurate results. A more detailed inspection, however, revealed that some of these models severely overfitted the experimental data, so not all models were considered acceptable. The best behavior was observed with the exponential models, which suggests that exponential decay processes are key elements of synapse activation. These results have several interesting implications, that can have an application in future neuroscience research. Firstly, as explained, we have identified the best possible candidate equations for both synaptic curves studied. Future studies of synaptic behavior can be modeled over these equations, further expanding the understanding of aspects such as synaptic geometry and neurotransmitter vesicle size. Our work demonstrates the validity of these equations. Secondly, we have identified the appropriate optimization techniques to be used when constructing these models. As we have demonstrated, not all curvefitting alternatives are adequate, so selecting the wrong one could have a strong impact on the validity of any derived research. Our work successfully resolves this issue, again setting the basis for upcoming advances in synaptic behavior analysis.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was financed by the Spanish Ministry of Economy (NAVAN project) and supported by the Cajal Blue Brain Project. The authors thankfully acknowledge the computer resources, technical expertise, and assistance provided by the Supercomputing and Visualization Center of Madrid (CeSViMa). A. LaTorre gratefully acknowledges the support of the Spanish Ministry of Science and Innovation (MICINN) for its funding throughout the Juan de la Cierva Program.