NEW VARIANTS OF THE DIFFERENTIAL EVOLUTION ALGORITHM: APPLICATION FOR NEUROSCIENTISTS L. Buhry, A. Giremus, E. Grivel, S. Sa¨ ıghi and S. Renaud IMS Laboratory, UMR 5218 CNRS University of Bordeaux – ENSEIRB 351, Cours de la Lib´ eration 33405 TALENCE – FRANCE Email: laure.buhry, audrey.giremus, eric.grivel, sylvain.saighi, sylvie.renaud@ims-bordeaux.fr ABSTRACT When dealing with non-linear estimation issues, metaheuristics are often used. In addition to genetic algorithms (GAs), simulating an- nealing (SA), etc., a great deal of interest has been paid to differen- tial evolution (DE). Although this algorithm requires less iterations than GAs or SA to solve optimization issues, its computational cost can still be reduced. Variants have been proposed but they do not necessarily converge to the global minimum. In this paper, our con- tribution is twofold: 1) we present new variants of DE. They have the advantage of converging faster than the standard DE algorithm while being robust to local minima. 2) To confirm the efficiency of our variants, we test them with a benchmark of functions often con- sidered when studying metaheuristic performance. Then, we use them in the field of neurosciences to estimate the parameters of the Hodgkin–Huxley neuronal activity model. 1. INTRODUCTION In the field of signal processing, several approaches operate in two steps: system modelling and model parameter estimation. The esti- mation step consists in searching for the set of parameters which minimizes a given error criterion such as the least square error, the maximum likelihood or the minimum variance. Depending on the constraints of the application (real-time and storage capacity), different off-line/on-line or iterative algorithms can be considered. Thus, one can use subspace methods, adaptive or optimal filtering, or the expectation-maximisation (EM) algorithm, etc. Their opti- mization steps may be based on classical techniques such as the steepest descent or the Newton Raphson methods. Nevertheless, when dealing with highly non-linear estimation issues, alternative solutions, such as metaheuristics, must be considered. They may be potentially useful, especially when locating the global optimum is a difficult task. Although their computational costs are quite high, they are well suited to optimization problems. These optimization techniques are inspired by natural systems like metallurgy for the simulated annealing (SA), biology of evolu- tion for the genetic algorithms (GAs) or ethology for the ant colony algorithms or theparticle swarm optimization. Thus, they have al- ready been used in a large number of application areas, such as biomedical applications to estimate intensity distributions for brain magnetic resonance images [11], Rayleigh fading channel simula- tion [5], in the field of multimedia [1], etc. In this paper, we focus our attention on the differential evo- lution (DE) algorithm. Invented by Price and Storn in 1995 [12], DE belongs to the class of evolutionary algorithms. Like GAs, DE consists of a population of individuals which evolves towards a pa- rameter vector that minimizes a beforehand defined fitness function F f it . The purpose of DE is then to explore and to evaluate new regions of the solution space, by building new candidate solutions from existing ones. It uses mechanisms inspired by biological evo- lution, namely the reproduction, the recombination, and the selec- tion. However, unlike GAs, the exploration is automatically regu- lated since new individuals result only from the recombination of individuals of the initial population. In addition, instead of keeping the best individuals of the k th generation 1 , DE consists in compar- ing pairwise one individual of a generation with its mutant. Thus, in DE, every candidate of the new generation has a fitness function F f it that is lower (or equal) to the fitness function of the previous one. Our contribution is twofold: firstly, we suggest a new variant of the DE in order to improve its convergence speed, hence its compu- tational cost, while avoiding local minima. Secondly, we evaluate the resulting approach in the field of neurosciences. Indeed, in this area, signal processing plays a role that is more and more active. Thus, we have already studied the relevance of GAs to estimate the parameters of the neuronal activity model proposed by Hodgkin– Huxley [6] and widely used in neurosciences [2]. The remainder of this paper is organized as follows. In the sec- ond part, we present the classical DE strategies. Then, in the third part, we describe the proposed variants and compare the perfor- mance of all the strategies on the benchmark of functions introduced by De Jong in [7] and widely used in the optimization field. Finally, in the fourth part, we apply and test our variant on a neuro–scientific problem: the parameter estimation of the Hodgkin-Huxley model. 2. CLASSICAL STRATEGIES OF THE DIFFERENTIAL EVOLUTION ALGORITHM 2.1 Classical DE DE consists in generating a population of NP 2 individuals which are composed of D parameters, also called “genes”. The population is initialized by randomly choosing individuals in a uniform manner within the boundary constraints of the model. Then, at each time step, new trial individuals are built by means of two operations: the so–called “differentiation” and the “recombination”. In the follow- ing, we define X r k (i) as the i th gene of the r th individual of the k th generation. Differentiation: the r th new parameter vector, X r k,trial , is gener- ated by adding to an individual X r 1 k randomly chosen among the k th generation in a uniform manner, the weighted difference between two other population members, X r 2 k and X r 3 k , with r 1 = r 2 = r 3 : ∀r = 1,..., NP , X r k,trial = X r 1 k + F.(X r 2 k − X r 3 k ), (1) where F is usually set to 0.5. Recombination: the r th mutant individual, X r k,mut , inherits genes of X r k,trial with a probability CR, where CR ∈ [0, 1] is usually set to 0.9. By generating u according to a uniform distribution over [0, 1] one has: ∀i = 1,..., D, ∀r = 1, .., NP , X r k,mut (i)=  X r k,trial (i) if u < CR X r k (i) otherwise. 1 k th “generation” denotes here the population at the iteration k. 2 For the sake of clarity, we use the same notations as in the seminal paper [12]. 17th European Signal Processing Conference (EUSIPCO 2009) Glasgow, Scotland, August 24-28, 2009 © EURASIP, 2009 2352