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Applying computers to problems in neuroscience that are unsolvable by experimental
methods.
by Laura Walbrink
Researchers of areas as diverse as circuit design and inventory control systems share a common problem: Devising a computational model to efficiently optimize the performance of high-dimensional non-linear systems. High-dimensional systems consist of a large number of variables (parameters). The vast number of possible combinations of the parameters within these systems renders the determination of an optimal parameter set impractical, if not impossible, by conventional means: enumerative, trial-and-error, hill-climbing, and calculus-based techniques.
The time needed to evaluate all potential sets of data in high-dimensional systems increases exponentially with the number of parameters. Consider, for ex-ample, a system containing 100 parameters. If each parameter had two possible states, and if each evaluation took just one second, evaluating all of the sets would take 4 * 1020 centuries (2100 seconds). Incidentally, the universe has only existed for approximately 2 * 108 centuries. Enumerative, or combinatorial, algorithms, used in low-dimensional systems but im-practical for high-dimensional systems, rely on testing each possible combination and are undesirable for that reason.
Trial-and-error, hill-climbing, and calculus-based techniques have also been used to solve optimization problems, but each method has drawbacks. With trial and error, a guess is made as to the correct parameter set based on experience with the system. The parameters are then manipulated after each evaluation until an acceptable combination is found. The trial-and-error method requires fewer evaluations than the enumerative approach. However, the optimal solution set may never be found.
The hill-climbing method begins with a set of random parameter values. While the other parameters remain constant, a "step" is taken along one dimension (parameter). The surrounding parameter sets are evaluated for optimal performance with the changed parameter. The set yielding the most improvement is chosen to become the new set of parameters. The process is repeated until no further improvement is achieved; the final set is the optimum. Hill climbing works best for parameters that are not highly interdependent. Unfortunately, this technique can become mired in local extrema and result in the discovery of an inferior solution. For this reason, several searches are conducted from unique random initial conditions, and their results are compared to verify that a global maximum has been found.
Calculus-based techniques search for the parameter value at which the solution manifold bifurcates. These routines, however, can only be applied to systems whose derivatives are both defined and continuous. In addition, the output might not correspond to the salient behaviors of the system. Efficiency is comparable to that of the enumerative method because software tools limit exploration to only one or two dimensions at a time.
The genetic algorithm, by comparison, is far more robust and efficient than any of the above methods. The method, based on the principles of natural selection, was conceived by John Holland in 1975. His Schema Theorem states that the genetic algorithm will consistently improve the overall fitness of an initially randomly chosen population of parameter sets.
The structure of the computational model is analogous to its biological counterpart. In the computational model, a chromosome is represented by an array of data which contain the parameters corresponding to the optimization problem. The chromosomes can be combined using techniques such as crossover and mutation. Crossover involves pairing selected parameters of one chromosome with others of a second chromosome. In small-bit mutation, one parameter is randomly selected and changed.
The efficiency of genetic algorithms is derived from their use of a random population base of relatively small size. Each individual combination is evaluated and assigned a fitness score to indicate its correlation to the optimal solution. In Darwinian fashion, an extinction threshold is applied to determine which individuals survive the generation. Those that live are combined with other survivors through crossover, mutation, or reproduction to produce the next generation. The unfit population is removed and the process repeated using the new generation. Each generation is increasingly fit and approaches the optimal solution.
Rogene Eichler West, a Ph.D. candidate in neuroscience, has applied the genetic algorithm to solve a problem currently irresolvable by experimental methods: extracting an accurate simulation of the behavior of a neuron from randomly selected potential data.
Proteins embedded in the cell's membrane serve as the non-linear control units that determine the neuron's response to incoming information. The conductances of the ion-permeable channels formed by the proteins vary with the voltage level and calcium concentration. Because ions pass through them into and out the cell, the channels themselves are the main source of changes in the calcium concentration and voltage level. If the voltage level rises above a certain threshold, an action potential (outgoing signal) transports information to neurons downstream; if the calcium concentration reaches a designated amount, biochemical changes occur. These changes are thought to comprise the molecular basis of learning and memory.
Neuroscientists discovered these channels over 40 years ago, and even more types have been identified recently due to advances in pharmacological tools and electrophysiological techniques. In some cases, the temporal behavior of these channels can be modeled through the derivation of kinetic equations. However, given a desired set of neuronal behaviors, the spatial distribution of channels with varied kinetics along the morphology of the neuron cannot be determined experimentally. Using Eichler West's genetic algorithm method, the distribution can be accurately predicted. Preliminary work involves experimentally obtaining and digitizing the three-dimensional morphology of a neuron. A partial differential equation based on this data is defined to model the flow of voltage and calcium within the cell. The morphology is then divided into six channel conductances: sodium, calcium, and four types of potassium. In a simplified model of a hippocampal neuron, 19 compartments were used for a total of 114 parameters.
At the initiation of the simulation, approximately five hundred individuals (parameter sets) are randomly generated from the pool of potential combinations. Each individual is run through the neuron simulator, a partial differential equation solver. The waveforms of the resultant time series, which describes five seconds of neuronal activity in response to an appropriate stimulus, are compared to experimentally obtained waveforms and assigned a fitness rating based on the degree of correspondence. The fitness score is sent to the genetic algorithm, which chooses the highest-scoring individuals to produce the next generation. Eventually, a generation is produced with a 100 percent fitness rate, and the optimal solution is achieved.
Eichler West uses the resources at the Laboratory for Computational Science and Engineering and the Minnesota Supercomputer Institute to evaluate this system through genetic algorithms. Because each parameter set can be simulated independently, the routines are suited to parallel processing. Supercomputers are employed to handle the magnitude of the problem. Even with the considerable computing power provided by a cluster of twelve Silicon Graphics R8000 processors, searching for the optimal set of parameters can require the simulation of more than 100,000 parameter sets and over a week of continuous processing. Running the same number of simulations on a Sun Sparc 2 processor would take more than a year.
Eichler West's work benefits the fields of neuroscience, control theory, self-organizing systems, and nonlinear optimization. The results of her work will support and highlight genetic algorithms as a viable technique for fitting nonlinear parameters of computational models to experimental data. The method is efficient, reliable, and can be modified to change the number of parameters, the computational model, or the fitness measure.
In the future, the computational model could serve as a reference or complement to animal-based experimentation. Well-defined neuronal mechanisms could inspire adaptive systems such as robots and pattern recognition devices. Eventually, it could provide the impetus for the next development in the 40-year-old single cell modeling theory.&
The Genetic Algorithm (GA) is a model of machine learning which derives its behavior from a metaphor of the processes of evolution in nature. This is done by the creation within a machine of a population of individuals represented by chromosomes, in essence a set of character strings that are analogous to the base-4 chromosomes that we see in our own DNA. The individuals in the population then go through a process of evolution.
