Firing properties of neurons are important in signal detection and generation in nervous system. Spinal cord motoneurons have membrane properties adapted to muscle properties. In this work we explored the relation between the first interspike interval and the amplitude of injected current. It was shown that increased sensitivity of firing frequency to the injected current at higher firing frequencies is due to the transient calcium current. Particularly, the specific density of N-type channels and the speed of their deactivation define the steepness and onset of the second range in frequency versus current relation

Svirskienė, N.

Svirskis, G.

English

Nonlinear Analysis: Modelling and Control

91Nonlinear Analysis: Modelling and Control, Vilnius, IMI, 2000, No 5Lithuanian Association of Nonlinear Analysts, 2000Firing Pattern Formation by Transient Calcium Currentin Model MotoneuronN. Svirskienė, G. SvirskisLaboratory of Neurophysiology, Kaunas Medical UniversityMickeviciaus 9, LT3000 Kaunas, Lithuaniagytis@cns.nyu.eduReceived: 09.07.2000Accepted: 27.09.2000AbstractFiring properties of neurons are important in signal detection and generationin nervous system. Spinal cord motoneurons have membrane propertiesadapted to muscle properties. In this work we explored the relation betweenthe first interspike interval and the amplitude of injected current. It was shownthat increased sensitivity of firing frequency to the injected current at higherfiring frequencies is due to the transient calcium current. Particularly, thespecific density of N-type channels and the speed of their deactivation definethe steepness and onset of the second range in frequency versus currentrelation.Key words: neural system, motoneuron, action potential, calcium channel,modeling.1 IntroductionInformation in nervous system is transmitted by series of actionpotentials, which are generated in neurons and targeted to other neurons orspecific organs. Although, it is not known how information is coded in theseseries of events, considerable efforts were applied to find out biophysicalmechanisms in neurons responsible for a generation of variety of action92potential patterns observed in behaving animals. Not surprisingly, it was foundthat neurons possess multiple types of ionic channels with different temporalproperties and sensitivity to membrane potential.Spinal cord motoneurons have obvious function to activate muscles,and, thus, are good objects to investigate how membrane properties are tuned bythe natural selection to neuronal function. Muscles by their own have properties,which are optimized for efficiency in producing movements. It is known thatoptimal force generation in fast muscles requires specific pattern of actionpotentials: if first several action potentials are fired with high frequency, a highforce could be further sustained by a following low frequency firing [10].On the other hand, intracellular recordings of membrane potential inmotoneurons showed that the frequency of firing depends nonlinearly oninjected current strength [7, 9]. This non-linearity manifests as two semi-linearranges where the second range is steeper than the first. Similar relations arepresent for the initial firing rate and for sustained stationary firing. The later wasinvestigated in modeling study previously [8] showing that slow calciumcurrents in dendrites are necessary for this phenomenon.In this work we investigated mechanisms of frequency to currentrelation (f-I) for the first two spikes. Modeling results suggest that transientcalcium current is responsible for the generation of the second steeper range off-I relation. The steepness is defined by the specific conductance of the current,and the point of the appearance is defined by the decay time constant.2 Model Definition and RealizationThe model neuron consisted of a spherical soma with the diameter of 100 µmand a dendrite. Dynamics of the soma membrane potential, V, were governed bythe following equation:CdV/dt = -V/Rm + Idendr  + I(V)S ,where C=0.16 pF is soma membrane capacitance, Rm=32 MΩ is membraneresistance, S is soma membrane area, Idendr is current flowing to the cablerepresenting dendritic tree. The passive membrane properties of the cable werethe same as in the soma, the length was 3mm, the diameter was 5 µm, andspecific intracellular resistance was 660 kΩ*µm. Idendr was calculated asdescribed previously [1]. I(V) is a current generated in 1 µm2 by four potentialdependent channels:93I(V) = GNaM3H (V - 55) + GKDr N3 (V + 85) + GCaMN2HN(V - 80) + GKCa(V ++85)CCa2/ (CCa2 + KCa),where GNa=140*10-5 µS/µm2, GKDr=80*10-5 µS/µm2, GCa, GKCa are specificconductances for sodium current, potassium delayed rectifier current, N-typecalcium current, and calcium sensitive potassium current respectively. CCa iscalcium concentration in µM. KCa=0.3 µM. Gating variables for the potentialdependent currents obeyed following relations:M = 1./(1. + exp[-(V + 35)/7.8]),dH/dt =  (1/(1 + exp[(V + 55)/7]) - H) / (30 / (exp[(V + 50)/15] + exp[ - (V ++50)/16])),dN/dt =  (1 / (1 + exp[ - (V + 20)/22.5]) - N) / (2/ (exp[(V + 40)/40] + exp[- (V ++40)/50])),dMN/dt = (1/ (1 + exp[ - (V + 30)/ 5]) - MN) / τact,dHN/dt = (1/ (1 + exp[(V + 45)/ 5]) - HN) / 12,where τact is activation time constant of N-type calcium current. The changes ofcalcium concentration were calculated by solving equationdCCa /dt = - KaICa - Kr (CCa - 0.05) ,where Ka =250 µM*µm2/nA, Kr=0.015 ms-1. The system of differentialequations was solved numerically by using implicit Cranck-Nicholson schemewith a time step of 5 µs.3 ResultsFirst, we defined parameter values, which resulted in action potentialand aftehyperpolarization reminiscent to that published in literature [6, 9].Because of nonlinear relations between parameters it would be difficult todefine a single set of parameters without a reasonable starting values. We tookthe initial values from a recent modeling study for motoneurons [3]. However, amodel with such parameters did not produced two different ranges in the f-I94relation of the first two spikes. Instead, the augmentation of the frequency wasless and less as the injected current value was increased.Fig. 1. Firing patterns of model motoneuron. A. Response to the 2.3 nA currentinjection. The specific conductance of calcium channels was 20*10-5 µS/µm2. Thecalcium current manifests as an afterdepolarization following fast hyperpolarization(arrows). B. Response to the 2.4 nA current injection. The afterdepolarization, hump,shown in A is responsible for the fast firing of the first two action potentialsIn our model with four potential dependent currents, two currents,0 50 100 150-80-60-40-200204060BAMembrane potencial, mVTime, ms0 50 100 150-80-60-40-200204060Membrane potencial, mVTime, ms95calcium and calcium sensitive potassium, could be responsible for the desiredeffect, since sodium current and delayed rectifier potassium current are toostrong and generate all-or-nothing action potential. We also noticed thatrecordings of membrane potential in motoneurons show a small “hump” justafter fast afterhyperpolarization [6]. To get the same shape ofafterhyperpolarization following action potential, the conductance of N-typecalcium channels, GCa, was increased. The stronger calcium current resulted inhigher calcium concentrations and, subsequently, in deeperafterhyperpolarizations. In order to compensate for this deepening, conductanceof the calcium sensitive potassium current, GKCa, had to be decreased. Also, thesensitivity of calcium dependent potassium current, KCa, had to be such that theincrease of calcium concentration after an action potential would not maximallyactivate this current. When KCa was equal to 0.3 µM the calcium dependentpotassium current was activated more with each next fired action potential. Forvalues of GCa=20*10-5 µS/µm2 and GKCa=5*10-5 µS/µm2 desired shape ofafterhyperpolarization with an initial hump was achieved (arrows in Fig. 1A).Next, we calculated f-I relation in this model. At small injected currentvalues the firing frequency increased almost linearly (Fig. 2A, dashed line), butat the current value of 2.4 nA the firing frequency of the first two spikes jumpedto high values. The mechanism of the jump could be understood by comparingfiring patterns just before (Fig 1A) and just after (Fig 1B) the jump. It can beseen that the second action potential moved to the “hump” resulting in the jumpof the firing frequency. However, in published experimental data such jumpsare absent.In order to make the transition between different f-I ranges smooth, wereduced calcium conductance to 10*10-5 µS/µm2 and adjusted conductance ofcalcium sensitive potassium conductance to 7*10-5 µS/µm2. In this case, theafterhypepolarization decayed almost without “humps” after the first spike, butit was still non-monotonous after the second spike resembling experimentallyobserved shape. These adjustments eliminated the jump in f-I relation (Fig 2A,solid curve) and made the second range of firing less steep. Next we asked whatparameter defines the onset of the second range. Apparently, it should be theduration of calcium current decay, since this current compensates for theoutward current and change the slope of afterhyperpolarization. To check this,the activation time constant, τact, which also sets the time scale of decay of thecurrent, was reduced from 12 ms to 8 ms. For such a model the onset of thesecond range in f-I relation moved to the higher frequencies as expected (Fig2B, dashed line). Also, the onset required stronger stimulation current, becausereduced activation time resulted in increased current, higher calciumconcentrations and deeper afterhyperpolarizations.96Fig. 2. Frequency-current relations for the first interspike interval. A. Comparison ofrelations between models with different strength of calcium current. The increase of thecalcium current resulted in steeper increase of the firing frequency. The activation timeconstant was 12 ms. B. When time constant for deactivation of calcium channel wasreduced the onset for the second range moved to higher firing frequencies. The specificconductance of calcium current was 10*10-5 µS/µm2.1.5 2.0 2.5 3.0050100 GCa=20 GCa=10Frequency, HzCurrent, nA2 3050100BA τact = 12 ms τact = 8 msFrequency, HzCurrent, nA974 ConclusionsIn this work we studied the mechanisms which are responsible for theshape of f-I relation for the first spikes in response to current injection inmotoneurons. We found that the second steeper range of f-I relation requiresstrong N-type calcium current in soma membrane. The beginning of this rangeis defined by the decay time constant of this current. These findings are inagreement with experimental observations. In cat motoneurons which havesimilar f-I relations and show stationary bistability [5, 4] the firing starts indoublets of spikes [2] similar to that in Fig 1B. Thus, transient calcium currentshape firing pattern in spinal motoneurons and enable them to effectivelyactivate their output targets, muscles.References1. A. Baginskas, A. Gutman, G. Svirskis, Bi-stable dendrite in constant electricfield: a model analysis. Neuroscience 53, p.595-603, 1993.2. D. J. Bennett, H. Hultborn, B. Fedirchuk, M. Gorassini, Short-term plasticityin hindlimb motoneurons of decerebrate cats. J.Neurophysiol. 80, p.2038-2045,1998.3. V. Booth, J. Rinzel, O. Kiehn, Compartmental model of vertebratemotoneurons for Ca2+-dependent spiking and plateau potentials underpharmacological treatment. J.Neurophysiol. 78, p.3371-3385, 1997.4. C. J. Heckman, R. H. Lee, The role of voltage-sensitive dendriticconductances in generating bistable firing patterns in motoneurons. J PhysiolParis 93, p. 97-100, 1999.5. J. Hounsgaard, H. Hultborn, B. Jespersen, O. Kiehn, Bistability of a-motoneurones in the decerebrate cat and in the acute spinal cat afterintravenous 5-hydroxytryptophan. J.Physiol.(Lond) 405, p. 345-367, 1988.6. J. Hounsgaard, O. Kiehn, I. Mintz, Response properties of motoneurones in aslice preparation of the turtle spinal cord. J.Physiol.(Lond) 398, p. 575-589,1988.7. D. Kernell, R. Bakels, J. C. V. M. Copray, Discharge properties ofmotoneurones: How are they matched to the properties and use of their muscleunits? Journal Of Physiology-Paris 93, p. 87-96, 1999.8. R. K. D. B. Powers, A variable-threshold motoneuron model thatincorporates time- and voltage-dependent potassium and calciumconductances. J.Neurophysiol. 70, p. 246-262, 1993.9. P. C. Schwindt, W. E. Crill, Factors influencing motoneuron rhytnmic firing:results from a voltage-clamp study. J.Neurophysiol. 48, p. 875-890, 1982.9810. C. K. Thomas, R. S. Johansson, B. Bigland-Ritchie, Pattern of pulses thatmaximize force output from single human thenar motor units. J. Neurophysiol.82, p. 3188-3195, 1999.

Firing Pattern Formation by Transient Calcium Current in Model Motoneuron

Firing properties of neurons are important in signal detection and generation  in nervous system. Spinal cord motoneurons have membrane properties  adapted to muscle properties. In this work we explored the relation between  the first interspike interval and the amplitude of injected current. It was shown  that increased sensitivity of firing frequency to the injected current at higher  firing frequencies is due to the transient calcium current. Particularly, the  specific density of N-type channels and the speed of their deactivation define  the steepness and onset of the second range in frequency versus current  relation

G. Svirskis

CiteSeerX

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.9.4028

Firing Pattern Formation by Transient Calcium Current in Model Motoneuron

Abstract

Similar works

Full text

Available Versions

Nonlinear Analysis: Modelling and Control

CiteSeerX