a genetic algorithm study on the influence of dendritic plateau potentials on bistable spiking in...

*Tel.: #973-596-5835; fax: #973-596-6467; e-mail: [email protected].

Neurocomputing 26}27 (1999) 69}78

A genetic algorithm study on the in#uence of dendriticplateau potentials on bistable spiking in motoneurons

Victoria Booth*Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights,

Newark, NJ 07102-1982, USA

Abstract

The necessity of dendritic plateau potentials for the generation of bistable spiking experi-mentally observed in vertebrate motoneurons was investigated in a minimal compartmentalmotoneuron model. Parameter regions where the model displays bistable behavior werecomputationally determined using genetic algorithms. Two "tness criteria were used to evalu-ate bistable behavior in the model: one using mathematically based techniques and the otherexperimentally based techniques. The performance of the GA depended on the "tness measureused but initial results con"rm that bistable spiking is robustly generated when spike-generat-ing currents are compartmentally segregated from plateau potential-generating currents withweak coupling conductance between the compartments. ( 1999 Elsevier Science B.V. Allrights reserved.

Keywords: Bistability; Motoneuron; Compartmental model; Genetic algorithm

1. Introduction

In serotonin (5-HT) and norepinephrine, cat spinal motoneurons display multiplebistable behaviors. For example, in response to a short-lasting excitatory input,a motoneuron that was initially at a resting potential can exhibit repetitive spikingthat persists long after the excitatory input has ceased (see [8] for review, [9]). Thisbistable behavior between a steady potential and repetitive "ring has been observed inother types of neurons and can be obtained with the space-clamped Hodgkin}Huxleymodel in a narrow parameter range. In 5-HT, cat motoneurons also display a more

0925-2312/99/$ } see front matter ( 1999 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 5 - 2 3 1 2 ( 9 9 ) 0 0 0 0 8 - 9

unusual bistable behavior whereby the cell can be shifted between two stable modes of"ring by short-lasting excitatory and inhibitory inputs [8]. These two spiking modescan di!er in frequency by as much as 10 Hz. This bistable spiking has not beenobserved in other types of neurons and is not easily obtained with standard, space-clamped neural models. The generation of these di!erent types of bistable behaviors isbelieved to depend on a persistent inward current at a dendritic location [8,9].

E!ects of 5-HT on spinal motoneurons were further investigated in a turtlepreparation where a persistent L-like Ca2` conductance capable of generating pla-teau potentials was identi"ed at dendritic locations [6]. The unmasking of thisconductance by 5-HT allowed the cell to display bistability between steady potentialsand repetitive "ring but bistable spiking was not readily observed.

To investigate the biophysical mechanisms underlying the generation of bistablespiking and speci"cally the role of dendritic conductances, two compartmental,motoneuron models were developed; an initial, minimal model [1] and a sub-sequent, more biophysically accurate model [2]. The results of both models con"rmedthat dendritic plateau potentials could account for the bistable "ring patterns ob-served experimentally. In addition, the modeling results suggested that obtainingall observed bistable and plateau behaviors may depend on a dendritic originfor the plateau potentials and may not be realizable in a single compartmentmodel.

These modeling results have prompted the following question: Is bistable spikinga signature behavior of dendritic plateau potentials? Such a categorization may bepossible since bistable spiking has special characteristics that di!erentiate it fromother bistable behaviors, namely two stable, repetitive "ring modes displaying spikesof similar amplitude but di!erent frequencies that can be observed over a largeapplied current range; and plateau potentials that are revealed when spike-generatingmechanisms are blocked. In this paper, we present initial results of our investigation ofthis question in which we identify parameter regimes where our minimal motoneuronmodel [1] displays any type of bistable behavior, but particularly where bistablespiking is observed. This search of parameter space is performed computationallyusing genetic algorithms (GAs) [5].

2. Model and genetic algorithm

The minimal motoneuron model [1] consists of two compartments, one represent-ing the soma and proximal dendrites and the second representing the lumped distaldendrites. Each compartment contains a minimal number of active currents, speci"-cally an inward (sodium- or calcium-mediated) and an outward (potassium-mediated)conductance, as well as a leakage current. The current-balance equations for eachcompartment are based on the dimensionless, Morris}Lecar equations. The compart-ments interact by current #ow through a coupling conductance, g

#. The coupling

conductance is modulated by a parameter p, representing the ratio of somatic surfacearea to total surface area. With the original parameter settings presented in [1], theisolated soma compartment displayed repetitive spiking for a range of applied current

70 V. Booth/Neurocomputing 26}27 (1999) 69}78

values, and the isolated dendrite compartment displayed plateau potential behavior.When g

#was set to small or moderate values, the coupled model displayed bistable

spiking. For large values of g#, however, bistable spiking was destroyed.

For the genetic algorithm parameter search, we varied the maximum conductancesof all currents, including leakage currents, in both compartments as well as thecoupling conductance, g

#, and the surface area ratio, p. The maximum conductances

were allowed to vary between 0 and 5 (dimensionless), g#

ranged between 0.1 and5 (dimensionless, the uncoupled case was not allowed), and p varied between 0.1 and0.9. The precision of all parameters was "xed at 0.1. All kinetic parameters for theconductance gating and time constants were set to the original values in [1]. With theoriginal maximum conductance values, these kinetic parameter values restricted spikegeneration to the soma compartment and plateau generation to the dendrite compart-ment. But over the range of maximum conductance values allowed in the search,spiking can occur in the dendrite compartment and plateau potentials in the somawith these same kinetic parameters.

Genetic algorithms have previously been applied to neural models with success[3,10]. Examples of other types of models where GAs have been successful are modelsof intracellular signalling by protein interactions [7] and models of underwateracoustics [4].

To represent our model parameters as chromosomes, we chose a real-valuedrepresentation over the usual binary representation. Each chromosome was a string ofeight real numbers consisting of three maximum conductances for each compartment(g

N!, g

K~$3and g

L4in the soma, and g

C!, g

Kand g

L$in the dendrite), the coupling

conductance g#

and the parameter p. Other GA studies have similarly chosen real-valued representations [3,7] to keep chromosomal length short and to promotesmooth convergence. In our GA, we kept a "xed population size of 500, allowing norepeated chromosomes within a generation. The initial population was randomlygenerated. At each subsequent generation, the 100 highest "t chromosomes wereautomatically placed in the next population and the remaining 400 new chromosomeswere created by recombination of members in the previous population. Pairs ofmating chromosomes were selected randomly with higher "t members having a higherprobability of selection. Recombination of chromosomes was performed by two-pointcrossover occuring with probability 1.0. Two mutation operators acted on the childchromosomes resulting from crossover. A local mutation operator perturbed a singleparameter in a chromosome by one precision unit ($0.1) with a 40% probability ofoccurrence. A global mutation operator replaced a single parameter with a randomlychosen value with a 10% probability of occurrence. To insure a thorough search ofparameter space, we performed "ve runs of the GA, each starting from di!erent initialpopulations and running for 50 generations.

We considered two di!erent "tness measures to evaluate bistable behavior in themodel. In each "tness function, the model was numerically integrated (using a variablestep-size, fourth-order Runge}Kutta scheme) at a series of somatic applied currentlevels (seven levels from !0.2 to 1.0). The "tness functions di!ered in the techniqueemployed to determine if the model showed bistable solutions at any of the appliedcurrent levels. One "tness function used the more mathematical or modeling-based

V. Booth/Neurocomputing 26}27 (1999) 69}78 71

Fig. 1. Bistable behavior in the model evaluated by the model-based "tness function (A,B) and theexperiment-based "tness function (C,D); (A,B) when integrated from di!erent initial conditions, modelsomatic (solid) and dendritic (dashed) voltages (dimensionless) display bistability between a steady potentialand repetitive "ring (A, somatic I

!11"0); and bistable spiking (B, I

!11"0.4); (C,D) in response to a somatic

applied current pulse (lower traces, magnitude 1), model voltages transition from a steady rest potential torepetitive spiking (C, holding I

!11"0); and between two stable "ring modes (D, holding I

!11"0.6).

technique of integrating the model from a series of di!erent initial conditions (eighttotal), at each applied current level. If the model behavior converged to di!erent stablesolutions, at the same applied current level, bistability was declared. Fig. 1 shows anexample of model performance with this "tness function. In panel A, the somaticapplied current level is held at 0. The top somatic and dendritic voltage trace showsthe model converging to a stable rest potential when one set of initial conditions is


used. In the bottom voltage trace, when a di!erent set of initial conditions are used,the model displays stable repetitive "ring. Thus, at zero somatic applied current, thisset of model parameters yield bistability between a steady rest potential and repetitivespiking. In panel B, when the somatic applied current level is set to 0.4, the samemodel parameter values show convergence to two di!erent, stable repetitive "ringmodes in response to di!erent initial conditions.

The second "tness function took the more experimentally based approach ofdiscovering bistable behaviors by applying two somatic current pulses of di!erentmagnitudes (0.5 and 1.0) from each in the series of holding current levels. Bistabilitywas declared if, following the current pulse, the model converged to a stable behaviordi!erent from the behavior observed before the pulse. Panels C and D in Fig. 1 showmodel performance to this "tness function at two di!erent holding current levels. Oneset of model parameters results in bistability between a steady rest potential andrepetitive spiking when somatic applied current was held at zero (C); and in bistablespiking at a higher holding applied current level (D).

In each "tness function, "tness points were incrementally awarded at each instanceof observed bistability. With each integration of the model during the "tness evalu-ation, a few points were added when the model converged to a recognizable behavior(#0.01 points for steady potentials and #0.03 points for singly-periodic, repetitive"ring). If the model behavior was not recognized, i.e. aperiodic, chaotic or non-convergent behavior, no "tness points were awarded and that model parameter setwas assumed non-physiological. Since we were primarily interested in the occurrenceof bistable spiking, the most "tness points were awarded for its observance (#0.06points). And since we wanted to focus on behaviors showing repetitive "ring, bistab-ility between a steady potential and repetitive "ring was more generously rewarded(#0.05 points) than bistable steady or plateau potential behaviors were (#0.04points). With this system of "tness point awarding, the maximum "tness valuepossible for the model is not known, however, the "tness performance of the originalmodel parameters (2.8 points with the model-based "tness function and 1.41 pointswith the experimentally-based "tness function) were used as a baseline to evaluate GAresults.

3. Results

3.1. Model-based xtness function

The "nal (generation 50) populations from all "ve runs of the GA were combinedand for each parameter varied, a "tness-weighted histogram was computed (Fig. 2).In a "tness-weighted histogram [4], instead of counting each occurrence of a para-meter value, the count is scaled by the "tness associated with the parameter set,thus only parameters in the maximum "tness parameter set receive a full count.This type of histogram is more suited to viewing GA results since there are still manylow "t chromosomes in the "nal population which may skew a standard histogramdisplay. Fig. 2 shows that, with the model-based "tness function, the GA converged


Fig. 2. Fitness-weighted histogram of combined "nal populations of "ve GA runs with the model-based"tness function. Solid vertical lines indicate the best representative parameter set (g

N!"3.7, g

K~$3"4.9,

gL4"0.7, g

C!"4.8, g

K"0.1, g

L$"2, g

#"0.2) and dotted vertical lines indicate original parameter values

(gN!"1, g

K~$3"2, g

L4"0.5, g

C!"1.5, g

K"0.5, g

L$"0.5, g

#"0.25).

to fairly narrow parameter ranges for some of the parameters varied (gN!

, gK~$3

, gL4

,gK

and g#) while a wide distribution of parameter values remained for g

C!and g

L$.

The GA also varied the parameter p, but its results are not included in Fig. 2 because,in the "nal populations, the distribution of values remained essentially uniform overthe interval [0.2, 0.7], indicating very little dependence on the value of p for bistablebehavior.

From the parameter values associated with the peaks in the histogram, a parameterset best representative of the GA results was chosen from the "nal populations (solidvertical lines, "tness "4.38). This parameter set, whose "tness performance is shownin Fig. 1A and B, yielded bistable spiking in the model at somatic applied currentlevels between 0.2 and 1.0, and bistability between a steady rest potential andrepetitive spiking at lower applied current levels ([!0.2, 0.0]). These bistablebehaviors were generated by the same model mechanisms as in the original modelalthough the parameter values were quite di!erent (compare with dotted verticallines in Fig. 2). Spike generation is limited to the soma compartment while thedendrite displays plateau potentials. Also, bistable behaviors are obtained forsmall values of the coupling conductance, g

#. In fact, the histogram peak

is the highest for g#

indicating strong convergence to a narrow range of g#

values(between 0.1 and 0.5).

Thus, the GA search of parameter space discovered mechanisms for the generationof bistable spiking similar to the mechanisms presented in the original model. Thesemechanisms are actually apparent from the histogram without looking at speci"cparameter sets. The relative values of g

N!, g

K~$3and g

L4indicate that the GA

converged to somatic conductances that generate repetitive spiking. In the dendrite


Fig. 3. Fitness-weighted histogram of combined "nal populations of "ve GA runs with the experiment-based "tness function. Solid vertical lines indicate the best representative parameter set (g

N!"4.8,

gK~$3

"0, gL4"3.9, g

C!"4.4, g

K"0.5, g

L$"1.5, g

#"0.3) and dashed vertical lines indicate maximum

"tness parameter values (gN!"2.3, g

K~$3"4.4, g

L4"1, g

C!"4.7, g

K"0.1, g

L$"1.7, g

#"0.5).

compartment, while the convergence was not as complete as with the soma conduc-tances, the high values for g

C!paired with much lower values of g

Kand a moderate

leakage gL$

indicate plateau potential behavior.

3.2. Experiment-based xtness function

Fig. 3 displays the "tness-weighted histogram for the combined "nal (generation 50)populations of the "ve GA runs with the experiment-based "tness function. It isimmediately apparent that convergence was not complete; wide ranges of valuesremain for all ionic conductances. However, the search process of the GA and thepromise of eventual convergence is re#ected in the histogram, particularly in themultiple peaks in the somatic active conductance values.

The best representative parameter set (solid vertical lines, "tness "2.13) yieldsplateau potential behavior in the model for a range of somatic applied current levels([0, 0.8]) with no repetitive "ring observed at any applied current level. This plateaupotential behavior is robustly achieved when both the soma and the dendrite arecapable of generating plateau potentials. The initial convergence of the GA to regionsof parameter space yielding this model behavior is evident by the peaks of g

N!and

gC!

at high values and the peaks of gK~$3

and gK

at low values. It is not surprising thatthe GA would locate this type of bistability initially since it is obtained over broaderparameter ranges than repetitive spiking.

The maximum "tness parameter set (dashed vertical lines, "tness"4.13), however,indicates that at the 50th generation the GA was shifting its convergence to parameterregions yielding bistable spiking. The model with the maximum "tness parametervalues shows bistability between a steady rest potential and repetitive spiking at low


somatic applied current levels ([!0.2, 0.2]) and bistable spiking at higher appliedcurrent levels ([0.4, 1.0]; see Fig. 1C and D). As in the results with the model-based"tness function, these bistable behaviors are achieved with spike generation restrictedto the soma compartment and plateau potential generation occurring in the dendritecompartment, and with the compartments weakly coupled. The convergence of theGA towards this region of parameter space is re#ected in the smaller peaks in thehistogram near these best "t parameter values. Additional GA runs continuing to 100generations con"rm the continued convergence to this region of parameter space(results not shown).

While complete convergence was not obtained in 50 generations of the GA with this"tness function, there was strong convergence to small values of the coupling conduc-tance g

#([0.1, 1.0]). The strong convergence to small g

#obtained with both "tness

functions re#ects the robustness of bistable behaviors when the compartments areweakly coupled.

4. Discussion

As our results testify, the design of the "tness measure is an important, andperhaps critical, element of the GA. For our investigation, we wanted to identifyparameter regions where the model showed any type of bistable behavior, includingplateau potential behavior, bistability between a steady potential and repetitivespiking, as well as bistable spiking. As our interest was in identifying where theseclasses of behaviors occurred, we did not specify, for example, values of restingpotentials or spike characteristics in our "tness measure. This broad design of the"tness measure results in a "tness landscape with plateaus, where ranges of para-meter values yield the same "tness value, rather than producing a "tness landscapewith high, isolated peaks and perhaps a single maximum "tness. A genetic algo-rithm is well-suited to this type of "tness landscape as the highest plateau can beidenti"ed and then the convergence of the population to the plateau can identifyits boundaries.

The convergence performance of the GA also critically depends on the "tnessmeasure used and its corresponding "tness landscape, as evidenced by the di!erencein convergence over 50 generations with our two "tness functions. The "tnesslandscape generated by the model-based "tness function surely contains largerparameter regions with high "tness than the landscape generated by the experiment-based "tness function. The main reason is that with the model-based "tness function,identifying bistability does not depend on transitioning between the two stablesolutions. Di!erent solutions are obtained by integrating from di!erent initial voltagelevels and gating variable values. With the experiment-based "tness function, on theother hand, the model must transition between the two stable solutions in response toa current pulse. This requires that the transition points or bifurcation points of thesolutions lie in a narrow range of somatic applied current levels (namely, between theholding current level and the pulse magnitude). This restriction narrows the regions ofparameter space where high "tness is obtained. In this case where the "tness landscape


has narrow, perhaps isolated, peaks the GA search will take more generations toconverge. This analysis of the "tness landscape suggests that bistability may occur inmany cell types but that the experimental protocols used to search for bistability maynot be able to discover it.

These initial results of GA identi"cation of model parameter ranges where bistablebehaviors are obtained con"rm that in our minimal motoneuron model with minimalcurrents in each compartment bistable spiking is most robustly obtained whenspike-generating mechanisms are compartmentally segregated from plateau-generat-ing mechanisms, and when the coupling between the compartments is weak. Thisproject is continuing using di!erent versions of the minimal model, including a one-compartment version with all currents in a single compartment and a two-compart-ment version with all currents in both compartments, as well as using our morebiophysical model [2], to investigate the dendritic in#uence on bistable spiking.

Acknowledgements

K. Sundaram for assistance in coding the GA and the following people for helpfuldiscussions on GAs: R. Eichler West, G. Wilcox, P. Rhodes, A. James, M. Recce, E.Michalopoulou and all members of the Mathematical Research Branch, NIH. Thisresearch was supported by the National Science Foundation (grant IBN-9722946).

References

[1] V. Booth, J. Rinzel, A minimal, compartmental model for a dendritic origin of bistability ofmotoneuron "ring patterns, J. Comput. Neurosci. 2 (1995) 299}312.

[2] V. Booth, J. Rinzel, O. Kiehn, Compartmental model of vertebrate motoneurons for Ca2`-dependentspiking and plateau potentials under pharmacological treatment, J. Neurophysiol. 78 (1997)3371}3385.

[3] R.M. Eichler-West, E. De Schutter, G.L. Wilcox, Using evolutionary algorithms to search for controlparameters in a nonlinear partial di!erential equation, in: Evolutionary Algorithms and HighPerformance Computing, IMA Volumes in Mathematics and its Applications, vol. 111, Springer, NewYork, 1998, in press.

[4] P. Gerstoft, Inversion of seismoacoustic data using genetic algorithms and a posteriori probabilitydistributions, J. Acoust. Soc. Am. 95 (1994) 770}782.

[5] D. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley,New York, 1989.

[6] J. Hounsgaard, O. Kiehn, Calcium spikes and calcium plateaux evoked by di!erential polarization indendrites of turtle motoneurones in vitro, J. Physiol. 468 (1993) 245}259.

[7] A. James, K. Swann, M. Recce, Cell behaviour as a dynamic attractor in the intracellular signallingsystem, J. Theoret. Biol. 196 (1999) 269.

[8] O. Kiehn, Plateau potentials and active integration in the "nal common pathway for motor behavior,TINS 14 (1991) 68}73.

[9] R.H. Lee, C.J. Heckman, Bistability in spinal motoneurons in vivo: systematic variations in rhythmic"ring patterns, J. Neurophysiol. 80 (1998) 572}582.

[10] M.C. Vanier, J.M. Bower, A comparison of automated parameter-searching methods for neuralmodels, in: J.M. Bower (Ed.), Computational Neuroscience: Trends in Research 1995, Academic Press,New York, 1996, pp. 477}482.


V. Booth received her Ph.D. in Applied Mathematics from Northwestern Univer-sity in 1993. From 1993}1996, she was a post-doctoral fellow in the MathematicalResearch Branch, NIDDK, NIH, working with J. Rinzel. Since 1996, she has beenan assistant professor in the Department of Mathematical Sciences at the NewJersey Institute of Technology.


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