Using Evolutionary Algorithms to Search for Control Parameters in a Nonlinear Partial Differential Equation

  • Rogene M. Eichler West
  • Erik De Schutter
  • George L. Wilcox
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 111)


Many physical systems of interest to scientists and engineers can be modeled using a partial differential equation extended along the dimensions of time and space. These equations are typically nonlinear with real-valued parameters that control the classes of behaviors that the model is able to produce. Unfortunately, these control parameters are often difficult to measure in the physical system. Consequently, the first task in developing a model is usually to search for appropriate parameter values. In a high dimensional system, this task potentially requires a prohibitive number of evaluations and it may be impossible or inappropriate to select a unique solution.

We have applied evolutionary algorithms (EAs) to the problem of parameter selection in models of biologically realistic neurons. Our objective was not to find the “best” solution, but rather we sought to produce the manifold of high fitness solutions that best accounts for biological variability. The search space was high dimensional (> 100) and each function evaluation required from one minute to several hours of CPU time on high performance computers. Using this model and our goals as an example, we will: 1) review the problem from the neuroscience perspective; 2) discuss high performance computing aspects of the problem; 3) examine the suitability of EAs for the efficient optimization of this class of problems; and 4) describe and justify the specific EA implementation used to solve this problem.


Genetic Algorithm High Performance Computing Recombination Operator Parallel Virtual Machine Soma Compartment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Abbott, L.F., Rolls, E.T., and Tovee,M.J., Representational capacity of face coding in monkeys,Cerebral Cortex 6: 498–505, 1996.CrossRefGoogle Scholar
  2. [2]
    Back, T., and Hoffmeister, F., Extended selection mechanisms in genetic algorithms, in Fourth International Conference on Genetic Algorithms in University of California, San Diego, edited by Belew, R.K., and Booker, L.B., Morgan Kaufmann, 92–99, 1991.Google Scholar
  3. [3]
    Back, T., Hoffmeister, F., and Schwefel, H.-P., A survey of evolution strategies, in Fourth International Conference on Genetic Algorithms in University of California, San Diego, edited by Belew, R.K., and Booker, L.B., Morgan Kaufmann, 2–9, 1991.Google Scholar
  4. [4]
    Bagchi, S. et al., Exploring problem-specific recombination operators for job shop scheduling in Fourth International Conference on Genetic Algorithms in University of California, San Diego, edited by Belew, R.K., and Booker, L.B., Morgan Kaufmann, 10–17, 1991.Google Scholar
  5. [5]
    Bhalla, U.S., and Bower, J.M., Exploring parameter space in detailed single neuron models: simulations of the mitral and granule cells of the olfactory bulb, Journal of Neurophysiology 69: 1948–1965, 1993.Google Scholar
  6. [6]
    Bramlette, M.F., Initialization, mutation and selection methods in genetic algorithms in Fourth International Conference on Genetic Algorithms in University of California, San Diego, edited by Belew, R.K., and Booker, L.B., Morgan Kauffman, 100–107, 1991.Google Scholar
  7. [7]
    Chetkovich, D.M. et al., N-Methyl-D-Aspartate receptor activation increases cAMP levels and voltage-gated Ca2+ channel activity in area CAl of hippocampus, Proceedings of the National Academy of Sciences of the USA 88: 6467–6471, 1991.CrossRefGoogle Scholar
  8. [8]
    Culberson, J., On the futility of blind search, University of Alberta Technical Report TR96–18, 1996.Google Scholar
  9. [9]
    Davis, L., Bit-climbing, representational bias, and test suite design, in Fourth International Conference on Genetic Algorithms in University of California, San Diego, edited by Belew, R.K., and Booker, L.B., Morgan Kaufmann, 18–23, 1991.Google Scholar
  10. [10]
    Davis, L. et al., A genetic algorithm for survivable network design in Fifth International Conference on Genetic Algorithms in University of Illinois at Urbana-Champaign, edited by Forrest, S., Morgan Kaufmann, 408–415,1993.Google Scholar
  11. [11]
    De Jong, K.A., An analysis of the behavior of a class of genetic adaptive algorithms, Doctoral Thesis, University of Michigan, Ann Arbor, 1975.Google Scholar
  12. [12]
    De Schutter, E., A consumer guide to neuronal modeling software. Trends in Neurosciences 15: 462–464, 1992.CrossRefGoogle Scholar
  13. [13]
    De Schutter, E., and Bower, J.M., An active membrane model of the cerebellar purkinje cell. I. Simulation of current clamps in slice, Journal of Neurophysiology 71: 375–400, 1994.Google Scholar
  14. [14]
    Denk, W., Strickler, J.H., and Webb, W.W., Photon laser scanning fluorescence microscopy, Science 248: 73–76, 1990.CrossRefGoogle Scholar
  15. [15]
    Ebner, T.J., and Chen, G.,Use of voltage-sensitive dyes and optical recordings in the central-nervous-system,Progress in Neurobiology 46: 463–506, 1995.CrossRefGoogle Scholar
  16. [16]
    Edmonds, B. et al., Contributions of two types of calcium channels to synaptic transmission and plasticity,Science 250: 1142–1146, 1990.CrossRefGoogle Scholar
  17. [17]
    Eichler West, R.M., On the development and interpretation of parameter manifolds for biophysically robust compartmental models of VA3 hippocampal neurons, Doctoral Thesis, University of Minnesota, 1996.Google Scholar
  18. [18]
    Eichler West, R.M., and Wilcox, G.L., A renumbering method to decrease matrix banding in equations describing branched neuron-like structures, Journal of Neuroscience Methods 68: 15–19, 1996.Google Scholar
  19. [19]
    Ferster, D., and Spruston, N., Cracking the neural code, Science 270: 756–757, 1995.CrossRefGoogle Scholar
  20. [20]
    Fitzgerald, K. et al., Multiple forms of non-associative plasticity in Aplysia: A behavioral, cellular, and pharmacological analysis, Philos Trans R Soc Lond 329: 171–178, 1990.CrossRefGoogle Scholar
  21. [21]
    Forrest, S., Genetic algorithms: principles of natural selection applied to computation, Science 261: 872–878, 1993.CrossRefGoogle Scholar
  22. [22]
    Foster, W.R., Ungar, L.H., and Schwaber, J.S,.J. S. Significance of conductances in Hodgkin-Huxley models, Journal of Neurophysiology 70: 2502–2518, 1993.Google Scholar
  23. [23]
    Geist, A. et al., PVM: Parallel Virtual Machine. A Users’ Guide and Tutorial for Networked Parallel Computing, Cambridge, MA: MIT Press, 1994.Google Scholar
  24. [24]
    Georgopoulos, A.P., Taira, M., and Lukashin, A., Cognitive neurophysiology of the motor cortex, Science 260: 47–52, 1993.CrossRefGoogle Scholar
  25. [25]
    Goldberg, D.E., Genetic algorithms in search, optimization, and machine learning, Reading, MA: Addison-Wesley, 1989.zbMATHGoogle Scholar
  26. [26]
    Goldberg, D.E., Real-coded genetic algorithms,virtual alphabets, and blocking, Complex Systems 5: 139–168, 1991.MathSciNetzbMATHGoogle Scholar
  27. [27]
    Goldman, D.E., Potential,impedance, and rectification in membranes,The Journal of General Physiology 27: 37–60, 1943.CrossRefGoogle Scholar
  28. [28]
    Hart, W.E., and Belew, R.K., Optimizing an arbitrary function is hard for Genetic Algorithms, in Fourth International Conference on Genetic Algorithms in University of California, San Diego, edited by Belew, R.K., and Booker, L.B., Morgan Kaufmann, 190–195, 1991.Google Scholar
  29. [29]
    Hille, B., Ionic Channels of Excitable Membranes, second ed., Sunderland, MA: Sinauer Associates Inc., 1992.Google Scholar
  30. [30]
    Hines, M., Efficient computation of branched nerve equations,International Journal of Biomedical Computing 15: 69–76, 1984.CrossRefGoogle Scholar
  31. [31]
    Hodgkin, A.L., and Huxley, A.F., A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology 117: 500–544, 1952.Google Scholar
  32. [32]
    Hodgkin, A.L., and Rushton, W.A.H., The electrical constants of a crustacean nerve fibre, Proceedings of the Royal Society of London Series B 133: 444–479, 1946.CrossRefGoogle Scholar
  33. [33]
    Holland, J.H., Adaptation in natural and artificial systems, Ann Arbor, MI: The University of Michigan Press, 1975.Google Scholar
  34. [34]
    Holmes, R.W., and Rall, W., Estimating the electrotonic structure of neurons with compartmental models, Journal of Neurophysiology 68: 1438–1452, 1992.Google Scholar
  35. [35]
    Jantsch, E., The self-organizing universe: scientific and human implications of the emerging paradigm of evolution,Elmsford, New York: Pergamon, 1980.Google Scholar
  36. [36]
    Johnston, D. et al., Active properties of neuronal dendrites, Annual Review of Neuroscience 19: 165–186, 1996.CrossRefGoogle Scholar
  37. [37]
    Kallen, R.G., Cohen, S.A., and Barchi, R.L., Structure, function, and expression of voltage-dependent sodium channels, Molecular Neurobiology 7: 383–428, 1993.CrossRefGoogle Scholar
  38. [38]
    Keynes, R.D., A new look at the mechanism of activation and inactivation of voltage-gated ion channels, Proceedings of the Royal Society of London Series B 249: 107–112, 1992.CrossRefGoogle Scholar
  39. [39]
    Kido, M. et al., Mantle viscosity derived by genetic algorithm using oceanic geoid and tomography for whole-mantle versus blocked-flow situations, Phys. Earth Planet Int., 1998.Google Scholar
  40. [40]
    Klein, M., and Kandel, E.R., Presynaptic modulation of voltage-dependent Cа2+ current: Mechanism for behavioral sensitization in Aplysia californica, Proceedings of the National Academy of Sciences of the USA 75: 3512–3516, 1978.CrossRefGoogle Scholar
  41. [41]
    Kuhar, M.J., and Unnerstall, J.R.,Quantitative receptor mapping by autoradiography: Some current technical problems, Trends in Neurosciences 49–53, 1985.Google Scholar
  42. [42]
    Kumar, V. et al., Introduction to Parallel Computing: Design and Analysis of Algorithms, Benjamin-Cummings Addison-Wesley Publishing Company, 1994.Google Scholar
  43. [43]
    Laurent, G., Dynamical representation of odors by oscillating and evolving neural assemblies, Trends in Neurosciences 19: 489–496, 1996.CrossRefGoogle Scholar
  44. [44]
    Li, M. et al., Convergent regulation of sodium channels by protein kinase C and cAMP-dependent protein kinase, Science 261: 1439–1442, 1993.CrossRefGoogle Scholar
  45. [45]
    Macready, W.G., Siapas, A.G., and Kauffman, S.A., Criticality and parallelism in combinatorial optimization,Science 261: 56–58, 1996.CrossRefGoogle Scholar
  46. [46]
    Maletic-Savatic, M., Lenn, N.J., and Trimmer, J.S., Differential spatiotemporal expression of K+ channel polypeptides in rat hippocampal neurons developing in situ and in vitro, Journal of Neuroscience 15: 3840–3851, 1995.Google Scholar
  47. [47]
    Mascagni, M.V., Numerical methods for neuronal modeling in Methods in Neuronal Modeling, edited by Koch, C., and Segev, I., Cambridge, Ma: MIT Press, 439–483, 1989.Google Scholar
  48. [48]
    Masukawa, L.M., Hansen, A.J., and Shepherd, G., Distribution of single-channel conductances in cultured rat hippocampal neurons, Cellular and Molecular Neurobiology 11: 231–243, 1991.CrossRefGoogle Scholar
  49. [49]
    Monster, A.W., and Chan, H., Isometric force production by motor units of exten sor digitorum communis in man, Journal of Neurophysiology 40: 1432–1443, 1977.Google Scholar
  50. [50]
    Numann, R., Caterall, W.A., and Scheuer, T., Functional modification of brain sodium channels by protein kinase C phosphorylation, Science 254: 115–118, 1991.CrossRefGoogle Scholar
  51. [51]
    Parsons, R.J., Forrest, S., and Burks, C., Genetic algorithms,operators, and DNA fragment assembly, Machine Learning 21: 11–33, 1995.Google Scholar
  52. [52]
    Perezreyes, E., and Schneider, T., Calcium channels - structure, function,and classification,Drug Development Research 33: 295–318, 1994.CrossRefGoogle Scholar
  53. [53]
    Press, W.H. et al., Numerical recipes, The art of scientific computing, second ed., Cambridge: Cambridge University Press, 1992.Google Scholar
  54. [54]
    Rall, W., Theory of physiological properties of dendrites, Annals New York Academy of Science 96: 1071–1092, 1962.CrossRefGoogle Scholar
  55. [55]
    Rall, W., Cable theory for dendritic neurons, in Methods in Neuronal Modeling, edited by Koch, C., and Segev, I., Cambridge, Mass: MIT Press, 9–62, 1989.Google Scholar
  56. [56]
    Rhodes, K.J. et al., Voltage-gated K+ channel beta-subunits - expression and distribution of КV-Beta-1 and KV-Beta-2 in adult rat brain,Journal of Neuroscience 16: 4846–4860, 1996.Google Scholar
  57. [57]
    Sakmann, B., and Neher, E., Single Channel Recording, New York: Plenum, 1983.Google Scholar
  58. [58]
    Saravanan, N., Fogel, D.B., and Nelson, K.M., A Comparison of Methods for Self-Adaptation in Evolutionary Algorithms,BioSystems 36: 157–166, 1995.CrossRefGoogle Scholar
  59. [59]
    Segev, I., Fleshman, J.W., and Burke, R.E., Compartmental models of complex neurons, in Methods in Neuronal Modeling, edited by Koch, C., and Segev, I., Cambridge, Mass: MIT Press, 63–96, 1989.Google Scholar
  60. [60]
    Snir, M. et al., MPI: The Complete Reference, Cambridge, MA: MIT Press, 1995.Google Scholar
  61. [61]
    Spears, W.M., Adapting Crossover in Evolutionary Algorithms, in Proceedings of the Fourth Annual Conference on Evolutionary Programming in San Diego, CA, 1991.Google Scholar
  62. [62]
    Spears, W.M., and De Jong, K.A., An analysis of multi-point crossover, in Proceedings of the Foundations of Genetic Algorithms Workshop in Bloomington, IN, 1990.Google Scholar
  63. [63]
    Spruston, N., Jaffe, D.B., and Johnston, D., Dendritic attenuation of synaptic potentials and currents - the role of passive membrane properties,Trends in Neurosciences 17: 161–166, 1994.CrossRefGoogle Scholar
  64. [64]
    Spruston, N., and Johnston, D., Perforated patch-clamp analysis of the passive membrane properties of three classes of hippocam pal neurons, Journal of Neurophysiology 67: 508–529, 1992.Google Scholar
  65. [65]
    Stuart, G.J., and Sakmann, B., Active propagation of somatic action potentials into neocortical pyramidal cell dendrites,Nature 367: 69–72, 1994.CrossRefGoogle Scholar
  66. [66]
    Syswerda, G., Uniform crossover in genetic algorithms, in Third International Conference on Genetic Algorithms, edited by Shaffer, J.D., Morgan Kaufmann, 1989.Google Scholar
  67. [67]
    Theunissen, F.E. et al., Information theoretic analysis of dynamical encoding by four identified primary sensory interneurons in the cricket cercal system, Journal of Neurophysiology 75: 1345–1364, 1996.Google Scholar
  68. [68]
    Toro, L., and Stefan, E., Calcium-activated K+ channels - metabolic regulation, Journal of Bioengineering-B 23: 561–576, 1991.Google Scholar
  69. [69]
    Traub, R.D. et al., Analysis of gamma rhythms in the rat hippocampus in vitro and in vivo, Journal of Physiology 493: 471–484, 1996.Google Scholar
  70. [70]
    Traub, R.D. et al., A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances, Journal of Neurophysiology 66: 635–650, 1991.Google Scholar
  71. [71]
    Turner, D.A., Segmental cable evaluation of somatic transients in hippocampal neurons (CAl, CA3, and Dentate), Biophysical Journal 46: 73–84, 1984.CrossRefGoogle Scholar
  72. [72]
    Turner, D.A., and Schwartzkroin, P.A., Steady-state electrotonic analysis of intracellularly stained hippocampal neurons, Journal of Neurophysiology 44: 184–199, 1980.Google Scholar
  73. [73]
    Westenbroek, R.E., Ahlijanian, M.K., and Catterall, W.A., Clustering of L-type calcium channels at the base of major dendrites in the hippocampal pyramidal neurons, Nature 347: 281–284, 1990.CrossRefGoogle Scholar
  74. [74]
    Whitley, D. et al., Comparing Heuristic,Evolutionary and Local Search Approaches to Scheduling, in Third Artificial Intelligence Planning Systems Conference, 1995.Google Scholar
  75. [75]
    Wolpert, D.H., and Macready, W.G., No free lunch theorems for search, Santa Fe Institute, 1995, 95–02–010.Google Scholar
  76. [76]
    Wonderlin, W.F., French, R.J., and Arispe, N.J., Recording and analysis of currents from single ion channels, In Neurophysiological Methods, edited by Vanderwolf, C.H., Clifton, NJ: Humana Press, 35–142, 1990.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Rogene M. Eichler West
    • 1
    • 2
    • 3
  • Erik De Schutter
    • 1
  • George L. Wilcox
    • 3
  1. 1.Born-Bunge FoundationUniversity of Antwerp — UIAAntwerpBelgium
  2. 2.Division of Biology, 216-76CaltechPasadenaUSA
  3. 3.Minnesota Supercomputer InstituteUniversity of Minnesota, MPLSUSA

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