Transition of Firing Patterns in a CA1 Pyramidal Neuron Model Dan Ma, Shenquan Liu, and Lei Wang
Abstract This paper considered a one-compartment and conductance-based model of CA1 pyramidal neuron. The InterSpike Interval(ISI) bifurcation diagrams were employed to demonstrate the transition modes of different firing patterns. The numerical results show that period-doubling and period-adding bifurcation phenomenon are exist in pyramidal cells in CA1 region.
1 Introduction Pyramidal neurons are a type of neuron found in areas of the brain including cerebral cortex, the hippocampus, and in the amygdala. The intrinsic discharge mode of individual cortical pyramidal cells varies along a spectrum of “burstiness” from regular firing evoked by depolarization of the neuron to spontaneous bursting unprovoked by any extrinsic stimuli [1, 2]. And a large body of evidence indicates that the propensity of a neuron to burst depends on a large number of internal and external factors, such as the type of ions channel, the ions concentration, the activation of ions channel, depolarizing currents and membrane capacitanceetc. Literatures about pyramidal neuron models and its behaviors are very rich, such as: Durstewitz et al. analyzed the irregular spiking in prefrontal cortex pyramidal neurons [3], Saraga et al. studied the spiking in the rodent hippocampus regulated by inhibitory synaptic plasticity [4], Royeck et al. made a good analysis of NaV1.6 sodium channels in action potential initiation of CA1 pyramidal neurons [5]. Nowadays, except for a larger number of neural electrophysiological experiments, there are many literatures which numerically simulated and analyzed the theoretical neuron models. Rinzel and Terman got abundant of firing patterns and the
D. Ma • S. Liu () • L. Wang Department of Mathematics, South China University of Technology, Guangzhou 510640, China e-mail:
[email protected] Y. Yamaguchi (ed.), Advances in Cognitive Neurodynamics (III), DOI 10.1007/978-94-007-4792-0 107, © Springer ScienceCBusiness Media Dordrecht 2013
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transmission modes between different neuron models [6, 7]. Izhikevich used the fast and slow bifurcation analysis to classify the various patterns of action potential bursting in detail [8]. The bifurcation patterns of neuronal firing sequences are also interested by other literatures. With the aid of InterSpike Intervals(ISI) bifurcation diagram, we give a detailed analysis of the variety of firing patterns in this model, and make a further analysis of period bifurcation phenomenon in the transitions of different firing patterns. The simulation results show the complex firing behaviors in CA1 pyramidal cells.
2 Model and Method The somatic, single-compartment model was represented by coupled differential equations according to the Hodgkin-Huxley-type scheme [9]. The model here was constructed in two stages. In the first stage, we introduced only the ionic currents that are involved in firing dynamics in zero [Ca2C ]o , at which it is simpler to analyze. In the second state, we added voltage-gated Ca2C and Ca2 C activated K C currents, to explore their influence on bursting behavior. In this article we only discuss the property of zero [Ca2C ]o . The model includes the currents that are known to exist in the soma and proximal dendrites: the transient sodium current (INa ) and the delayed rectifier potassium current (IKdr ) that generate spikes, and the muscarinic-sensitive potassium current (IM ) that contributes the slow variable necessary for bursting [10, 11]. A model with these three currents only is the minimal model that allows bursting. We added the persistent sodium current (INaP ) because we wanted to focus on its contribution to bursting [1]. The A-type potassium current (IA ) and the leak current (IL ) are included as well. The kinetics equations and parameters are listed in Table 1. The cell model for zero [Ca2C ]o has five dynamical variables: V; h; n; b and z. Table 1 Kinetics equations and parameters for the zero [Ca2C ]o model Current, variable INa ; m INa ; h
INaP ; p IK dr ; n
IA ; a IA ; b IM ; z
Kinetics/Time Constant, ms
Parameters
m D m1 .V / dh=dt D k Œh1 .V / h =k .V / k .V / D 0:1 C 0:75 f1 C exp Œ .V k2 / =k2 g1 p D p1 .V / d n=dt D n Œn1 .V / n =n .V / n .V / D 0:1 C 0:5 f1 C exp Œ .V n2 / =n2 g1
m D 30 mV; m D 10:5 mV k D 45 mV; k D 7 mV k2 D 40:5 mV; k2 D 6 mV; k D 1 p D 45 mV; p D 3 mV n D 35 mV; n D 10 mV n2 D 27 mV; n2 D 15 mV; n D 1
a D a1 .V / db=dt D Œb1 .V / b =b ;b D 15 d z=dt D Œz1 .V / z =z ; z D 75
a D 50 mV; a D 20 mV b D 80 mV; b D 6 mV z D 39 mV; z D 5 mV
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The current balance equation is: dV D I L INa INaP IKdr IA IM C Iapp (1) dt ı ı where C D 1F cm2; gL D 0:05ms cm2; VL D 70mV; and Iapp is the applied current. The ionic currents are: C
INa .V; h/ D gNa m1 3 .V /h.V VNa /; IKdr .V; n/ D gKdr n4 .V VK /; Im .V; z/ D gM z.V VK /; IA D gA a1 3 .V /b.V VK / INaP .V / D gNaP p1 .V /.V VNa /; IL D gL .V VL / The activation and inactivation curves x1 .V / are determined by the equation x1 .V / D f1 C expŒ.V x / =x g1 . Where x D m; h; p; n; a; b; z:
3 Simulation Results and Analysis 3.1 The Influence of Membrane Capacitance and External Stimulus Since neuron membranes are composed of double- membrane phospholipid, the thickness and cytoplasm component are similar, so difference of C for different neurons is not large, but the little distinction always exists as to a variety of neurons. Thus in this section we give an analysis of the influence of C to the generation and conduction of neural action potential sequences and obtains various bifurcation patterns in the model. See Fig. 1a, b for details. External stimulus is an important factor which can influence neuronal firing activities obviously. See Fig. 1c for more information. From diagram a b we can see that with the increasing of C, the firing pattern of CA1 pyramidal neuron changes from period firing modes to period complex bursting, and the spikes in the bursting are increasing too. The firing sequences show a clear period adding phenomenon. In diagram b, with the increasing of C from 0.7 to 1.2, ISIs sequence changes from period-1 firing doubling to period-2 bursting, continues after another period doubling bifurcation turns into period-4 bursting, then turns into period-3 bursting through a complex region, after that turns into period adding bifurcation phenomenon.
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Fig. 1 Firing patterns in the model. (a) Under the stimulation of Iapp D 1:0nA, period adding firing sequences of membrane potential from period-1 to 5 for C D 0.8, 0.9, 0.93, 0.95, 1.0 and 1.05 respectively. (b) Corresponds to diagram a, the bifurcation diagram of C vs. ISIs. (c) The bifurcation diagram of Iapp vs. ISIs, the diagram on the top right corner is the enlargement of diagram c
From the analysis of diagram c, we know that with the decreasing of dc stimulus from 2 to 0.4, the ISIs sequence of this neuron model change from period-1 firing doubling to period-2 bursting, continue after two period doubling bifurcations reach to period-8 bursting, then turn into period-3 bursting through a complex region.
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3.2 The Influence of the Transient Sodium Current In the description of the classical HH model, the change of sodium ion conductance and potassium ion conductance are determined by three variables: m, h and n, where the change of m and h determine the amplitude of sodium ion current, and the change of n determine the amplitude of potassium ion current. The change speed of these variables determine ion channel current’s amplitude at different time, and then influence the firing pattern of action potential, so the perturbation of ion channel variable is critical in the activities of neuron. In this part, we make a simple discussion about the variables m, h of the transient sodium current. See Fig. 2a, b, c for details. At present, there is still a lot of literature in which the influence of ions channel conductance and ions equilibrium potential to neuron activity were investigated. In this section, by changing the ion conductance and ion equilibrium potential of transient sodium current, rich bursting patterns and the bifurcation phenomenon in the switching between different patterns of firing are obtained. See Fig. 2d. From the above figures we can get several bifurcation patterns in neuronal action potential ISIs. In diagram a b, the sequences of action potential ISIs show a clear bifurcation phenomenon with the perturbation of ion channel variable m. Both diagram a and b contain a clear period doubling phenomenon, a clear period adding phenomenon and a clear inverse period doubling phenomenon. In diagram c, we can see the period doubling bifurcation and period adding bifurcation clearly. So we know only a little change of the perturbation variables m and h of the transient sodium current can change the firing modes of CA1 pyramidal neuron easily. And the ISIs sequences have good bifurcations. Diagram d shows the influence of the ion conductance of transient sodium current to neuron activity. With the decreasing of gNa from 58 to 46, the ISIs sequence of this neuron model appears three period doubling bifurcations, and then turns into a complex region. The numerical results above show that the perturbation variables and ion conductance of transient sodium current have great influence on neuronal activities (action potential). These rich bursting patterns and the switching between different patterns of firing tell us that transient sodium current is very important in the generation of the rich bursting.
4 Discussions From the analysis of simulation results in this paper we can find that the firing modes in zero [Ca2C]o model of CA1 pyramidal neuron are very rich. And it allowed us to use a single-compartment model to analyze the underlying mechanism of bursting. The model results above tell us that the CA1 pyramidal neuron can show a variety of firing patterns when the electrophysiology parameters vary in a certain region, not only periodic spiking but also various bursting patterns. The ISI bifurcation
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Fig. 2 Bifurcation diagrams of neuronal action potential ISIs. (a) ISIs vs. m . (b) ISIs vs. m . (c) ISIs vs. (h D n ). (d) ISIs vs. gNa
phenomenon we drew can show more intuitive results of how these firing patterns change from one pattern to another, the period doubling phenomenon and the period adding phenomenon contained in the ISI bifurcation diagrams tell us the intrinsic properties of corresponding parameters on the spiking transition modes and firing
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patterns in the CA1 pyramidal neuron. In fact many others parameters in this model can affect the fire patterns transition, such as the delayed rectifier potassium current, A-type potassium current and A-type potassium current et al and all these currents change demonstrate the basic rule of neuron codes. Acknowledgments This work is supported by the National Nature Science Foundation of China under Grant Nos. 10872069.
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