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From the early days of complex systems research, 40's and 50's, biological systems were considered as an inspiration for the design of complex artificial systems. But of course, understanding the self organisation properties of biological systems was itselfa major challenge.

We will not describe here in details the bio-inspired applications such asthe many applications of neural nets research to signal processing: the reader might check Mark Buchanan document, or the standard literature on neural nets. We similarly skip DNA computation, immuno-computing etc.

We will rather concentrate on complex system research motivated by the understanding of biological systems such as cognition, genome expression,the immune system and ecology.

This basic research, often referred to assystem biology, poses fundamentalscientific questions; furthermore, its application to health,medicine, drug design, environmental issues is of the highest importance. For instance abetter understandingof the interactions that control gene expression would considerably facilitate the design of new drugs,both in terms inventing the drug and synthetising it, but it would also in helping to check for possible negative side effects.

Year 2000 has beena turning point in the application of complex system research to biology.

Before 2000 the major effort was the understanding of functional properties of biological organs: for instance how can neuron assemblies give rise to cognitive properties in the brain? Hopfield 1982 PNAS paper on associative memories of assemblies of formal neurons is a typical example. Hopfield demonstrated formally that sets of formal neurons educated by a simple delocalised algorithm, Hebbs rule, could later recognise the patterns they had been teached; this recognition does not evenrequesta presentation of the original pattern: even presentation of a part of the pattern would suffice. Most of the research of the 40's to year 2000 was long these lines: the cellular automata of von Neumann were proposed as models of self-reproduction, S. Kauffman (1969) modeled phenotypic expression based on interactions among gene expression, R. Thomas, M. Kaufmann, A. S. Perelson etal from the 70's modeled the immune response as a result of idiotypic interactions among antibodies and cell receptors.

A large fraction of the research interpreting the origin of life (Eigen 72, P. W. Anderson 83, S. Kauffman 89) as due to the emergenceof a set of autocatalytic polymers belong to this line of research.

At that time, the complex system approach was an alternative to thereductionist approach which considered biological functions as the result of a finely tuned selection and proneda major effort at specifying the details of all involved components and of their interactions. The complex systemscientists established that functional properties could be the result of interactions among rather simple entities. Functional organization was shown to be a generic property of the attractors of the dynamics of the interacting system rather than the result highly tuned natural selection.

These earlier efforts greatly improved our insightin biological organisation.

The availability of huge sets of data since the 90's: transcriptomes obtained from biochips, multi-neuron recording, new techniques for imaging plus what we canexpect from the near future in proteomics for instance, open a new era in complex systems methods.

Can we meet the challenge of this data driven biologyand extract sensible information from these very large data sets?

To be more specific can we for instance extractthe specific sets of interactions among genes which control gene expression in the cell? In mathematics such a problem is called an inverse problem: rather than finding a trajectory knowing the dynamics governing the system, we here have to solve the inverse problem, getting the dynamics from the observation of successive configurations of expressed genes.

Inverse problems are known to be much harder to solve than direct problems. Furthermore, in system biology one hasto solve a problem with a very large number of unknown parameters.

Even when supposing that the set of interactions among an ensemble of N elements is sparse rather than complete, we obtain a numberof parameters to determine scaling as N (N^2 in the case of complete connectivity). Furthermore, in general, the structure of the network is a priori unknown, and so is the nature of the interaction function.

Fortunately, inverse problems relate to a number of problems already treated in complex system research.

Learning by formal neuron networks is an inverse problem: it simply means finding the set of interactions which for instance associates ouptut patterns to input patterns. Variants includepattern sequence generators. An exciting prospect is to use and generalise learning techniques such as the perceptron algorithm to e.g. decoding gene regulatory networks using transcriptome data. This approach suppose thatthe function regulating the expression of each gene arethreshold automata, a not unreasonable assumption, but other techniques inspired from complex system research can be used to solve this problem. An alternative approach is to used the survey propagation algorithm developed for constraint satisfaction problems.

A very general, but more time consuming method is based on the use of genetic algorithm.

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