Plant model

The Plant model is one of the first conductance-based models developed to explain the mechanism underlying parabolic bursting oscillations observed in the membrane potential of the R15 pacemaker neuron from the abdominal ganglion of the mollusk Aplysia.

Motivation

In the early 1970's, the basis for all meaningful modeling of nerve membrane behavior was the Hodgkin-Huxley model for action potentials and repetitive firing.

The motivation for the development of the Plant model was to create a mathematical model that exhibited more complex behavior consistent with experimental observations. Arguably, the simplest behavior more complex than a steady stream of action potentials is bursting. At that time, the best known experimental preparation displaying bursting behavior was the cell designated R15 of the mollusk Aplysia, extensively studied by a number of researchers, including Frazier et al (1967), Junge and Stevens (1973), Mathieu and Roberge (1971), and Strumwasser (1971). The two following experimental observations were important in the development of the Plant model and its forerunners:

• The bursting phenomenon observed in the R15 cell is not related to traveling wave properties of the action potential; it can be observed in a spatially uniform nerve membrane.
• When the poison tetrodotoxin (TTX) of the puffer fish is applied to the membrane of the R15 cell, the action potentials disappear and an underlying slow, periodic membrane oscillation emerges.

Model Equations

The model now known as the Plant model was published by Plant in 1981. It was informed by earlier modelling efforts which began in 1975 (see Earlier Versions of the Model, below).

The model is based on the following current balance equation (variable names have been changed to conform to the more conventional standard from the well-known Hodgkin-Huxley model): \[ C_m \frac{dV}{dt} = - \bar{g}_I \cdot m^3_{\infty}(V) \cdot h \cdot (V-V_I) - \bar{g}_T \cdot x \cdot (V-V_I) - \bar{g}_K \cdot n^4 \cdot (V-V_K) + \bar{g}_{KCa} \cdot \frac{ Ca }{ 0.5 + Ca} \cdot (V-V_K) - \bar{g}_L \cdot (V-V_L), \] where \(V\) is the membrane potential (mV) and \(t\) is time (ms). \(C_m\) is the membrane capacitance (\(\mu\)F/cm²), \(\bar{g}_i\) are maximal ion conductances (mmho/cm²), \(V_i\) are Nernst reversal potentials (mV), \(m_{\infty}(V)\) is the instantaneous activation function for the fast inward current, \(x\) and \(n\) are dimensionless activation variables, \(h\) is a dimensionless inactivation variable, and \(Ca\) represents the dimensionless calcium ion concentration at the inner face of the membrane (more generally, it represents the intracellular calcium concentration).

• The first and third ion currents in this equation essentially are the two classical Hodgkin-Huxley currents, responsible for generating action potentials, with minor modifications. In particular, the inward current is assumed to have instantaneous activation, and assumed to be a combined inward current due to both sodium and calcium (\(V_I\) thus is a mixed effective Nernst reversal potential).
• The second and fourth ion currents operate on much slower time scales, and are responsible for the slow, periodic oscillation that drives the bursting phenomenon. In particular, the second current is a TTX-resistant inward current with slow activation, and the fourth current is an outward current carried by potassium ions, sensitive to an assumed slow buildup of intracellular calcium ions.
• The fifth ion current is a generic inward leak current.

Figure 1: Upper panel: Bursting exhibited by the isopotential Plant model, with parameter values as specified in the Plant (1981) paper. Middle panel: Slow oscillations in the membrane potential in the presence of TTX, simulated by setting \(\bar{g}_I=0\ .\) Lower panel: Bursting with a parabolic profile in interspike intervals, obtained by setting \(\bar{g}_I=4\ ,\) \(\rho=0.00015\ ,\) \(K_c=0.00425\ ,\) \(\tau_x=9400\ ,\) and \(x_{\infty}=\left[ \exp(-0.3(V+40)) + 1 \right]^{-1}\ .\)

The equations for the activation and inactivation variables are as follows: \[ \begin{array}{lcl} \frac{dh}{dt} & = & \frac{ h_{\infty}(V) - h }{ \tau_h(V) }, \\ \frac{dn}{dt} & = & \frac{ n_{\infty}(V) - n }{ \tau_n(V) }, \\ \frac{dx}{dt} & = & \frac{ x_{\infty}(V) - x }{ \tau_x }, \end{array} \] and the equation for the intracellular calcium concentration is: \[ \frac{dCa}{dt} = \rho \left[ K_c \cdot x \cdot (V_{Ca}-V) - Ca \right]. \] The steady-state activation and inactivation functions are given by \[ \begin{array}{lcl} w_{\infty}(V) & = & \frac{ \alpha_w(V) }{ \alpha_w(V) + \beta_w(V) } \mbox{ for } w = m, h, n, \\ \tau_w(V) & = & \frac{ 12.5 }{ \alpha_w(V) + \beta_w(V) } \mbox { for } w = h, n, \\ \end{array} \] and \[ x_{\infty}(V) = \frac{1}{ \exp(-0.15(V+50)) + 1 }, \] where \[ \begin{array}{lcl} \alpha_m(V) & = & 0.1 \frac{ 50-V_s }{ \exp((50-V_s)/10) - 1 }, \\ \beta_m(V) & = & 4 \exp((25-V_s)/18), \\ \alpha_h(V) & = & 0.07 \exp((25-V_s)/20), \\ \beta_h(V) & = & \frac{ 1 }{ \exp((55-V_s)/10) + 1 }, \\ \alpha_n(V) & = & 0.01 \frac{ 55-V_s }{ \exp((55-V_s)/10) - 1 }, \\ \beta_n(V) & = & 0.125 \exp((45-V_s)/80), \end{array} \] and \[ V_s = \frac{127}{105} V + \frac{8265}{105} \] (not \(V_s = \frac{127}{105} V - \frac{8265}{105}\) as stated in the original paper).

The value of the membrane capacitance is \(C_m = 1 \mu\)F/cm². The values of the maximal conductances are \(\bar{g}_I = 4.0\ ,\) \(\bar{g}_T = 0.01\ ,\) \(\bar{g}_K = 0.3\ ,\) \(\bar{g}_{KCa} = 0.03\ ,\) and \(\bar{g}_L = 0.003\ ,\) all with the units of mmho/cm². The value of the Nernst reversal potentials are \(V_I = 30\ ,\) \(V_K = -75\ ,\) \(V_L = -40\ ,\) and \(V_{Ca} = 140\ ,\) all with the units of mV. The values of the remaining parameters are \(\rho = 0.0003\) ms^-1, \(K_c = 0.0085\) mV^-1, and \(\tau_x = 235\) ms.

Behavior of the Model

• A simulation of the full isopotential Plant model, using the parameter values as specified in the Plant (1981) paper, results in bursting, as shown in the upper panel of Figure 1.
• The application of the poison TTX is simulated by setting the conductance of the TTX-sensitive inward current, \(\bar{g}_I\ ,\) to zero. A slow, periodic oscillation of the membrane potential emerges, as shown in the middle panel of Figure 1.
• A graph of the interspike intervals versus spike number within each burst of action potentials does not show the parabolic shape characteristic of parabolic bursting. However, the model can yield parabolic bursting after changing the values of the model parameters, as shown in the lower panel of Figure 1.

Dissection of the Model

Figure 2: Upper panel: One-parameter bifurcation diagram of the fast subsystem of the Plant model, with \(x=0.9\ .\) Thin black curves represent the branch of steady states (solid = stable steady states; dotted = unstable steady states) and thick black curves represent the maximum and minimum of the branch of periodic solutions (solid = stable oscillations; dotted = unstable oscillations). HB indicates a Hopf bifurcation; SNIC the saddle-node on invariant circle bifurcation. Lower panel: Corresponding two-parameter bifurcation diagram, showing the continuation of the Hopf bifurcation and the saddle-node on invariant circle bifurcations. In the region immediately to the left of the curve of SNICs, the fast subsystem is oscillatory; to the right, the fast subsystem exhibits a steady state. Superimposed is the projection of the bursting solution from the upper panel of Figure 1, showing that the solution of the full Plant model periodically visits the oscillatory and steady-state regions, resulting in bursting.

• Rinzel and Lee (1986, 1987) analyzed the Plant model to uncover the mathematical mechanism underlying parabolic bursting by decomposing the model into a fast and a slow subsystem, based on the separation of time scales on which the ion currents operate.
• The fast subsystem of the Plant model consists of the equations for membrane potential \(V\ ,\) and the (in)activation variables \(h\) and \(n\ ;\) the slow subsystem consists of the equations for the activation variable \(x\) and the intracellular calcium concentration \(Ca\ .\)
• The key feature of the fast subsystem of the Plant model is that its bifurcation diagram (obtained with the slow variable \(x\) held constant and the slow variable \(Ca\) used as the bifurcation parameter) contains a branch of steady states (corresponding to the quiescent periods between bursts) and a branch of periodic solutions (corresponding to the action potentials within bursts) connected via a saddle-node on invariant circle bifurcation, as shown in the upper panel of Figure 2.
• Bursting is obtained from an oscillation in the slow system that periodically moves the \(Ca\) variable back and forth across the saddle-node on invariant circle bifurcation, from the branch of steady states to the branch of periodic solutions and vice versa, as shown in the lower panel of Figure 2.
• Since the transition between the active and quiescent phases of bursting involves passing through a saddle-node on invariant circle bifurcation, which is associated with a limit cycle with infinite period, the spike frequency is low at the beginning of the burst, then increases, and then decreases again towards the end of the burst. Hence the graph of the interspike interval versus spike number within each burst of action potentials is parabolic.
• Following the dissection of the Plant model by Rinzel and Lee and subsequent classification schemes, the bursting mechanism underlying the Plant model is known equivalently as a parabolic burster (Rinzel (1987)), a type II burster (Bertram et al. (1995)), and a burster of circle/circle type (Izhikevich (2000)).

Significance of the Model

• The Plant model and its forerunners were among the first (if not the first) conductance-based models of bursting consistent with experimental observations, in particular parabolic bursting as observed in the Aplysia R15 cell. The development and refinement of conductance-based models of bursting phenomena continues to this day.
• The model stimulated mathematical investigation. The successful dissection of the Plant model by Rinzel and Lee (1986, 1987) contributed to a deep understanding of bursting mechanisms and the popularization of qualitative or geometric perturbation analysis.
• Rinzel and Lee (1986, 1987) showed that the mechanism underlying bursting in models for the R15 cell differs from the mechanism in models of bursting for the pancreatic beta cell. This finding was the impetus for a description of different classification schemes of bursters (Rinzel (1987), Bertram et al (1995), de Vries (1998), Izhikevich (2000), and Golubitsky et al (2001)).

Earlier Versions of the Model

The forerunners of the Plant model were the models presented in Plant and Kim (1975), Plant and Kim (1976), and Plant (1978). The evolution of the models reflects the evolution in the hypotheses, informed by experimental observations, about the ion channels thought to play an important role in the generation of the bursting phenomenon.

Plant and Kim, 1975

The model by Plant and Kim (1975) is based on the following current balance equation: \[ C_m \frac{dV}{dt} = - \bar{g}_I \cdot x^3_I \cdot y_I \cdot (V-V_I) - \bar{g}_K \cdot x^3_K \cdot (V-V_K) - \bar{g}_A \cdot x^4_A \cdot y_A \cdot (V-V_K) - \bar{g}_P \cdot x_P \cdot (V-V_K) - \bar{g}_L \cdot (V-V_L) + I_P, \] where \(x_i\) are the activation variables, \(y_i\) are the inactivation variables, \(I_P\) represents a tonic electrogenic pump current, and all other symbols are as described above.

Here, the action potentials are generated by the classical Hodgkin-Huxley currents (the first and second ion currents in the above equation), and the slow, periodic oscillation observed in the presence of TTX is generated by two additional potassium currents (the third and fourth currents in the above equation) based on the work of Connor and Stevens (1971). One is a potassium current with both activation and inactivation, and the second is a potassium current with very slow activation. The model is completed with the specification of the dynamics for the activation and inactivation variables \(x_i\) and \(y_i\ .\) See Plant and Kim (1975) for details.

Of historical interest may be the observation that Plant and Kim did not include a numerical solution of their model, reflecting the state of computing at that time.

Plant and Kim, 1976

In 1976, Plant and Kim published an updated version of their model (including a numerical solution of their model). Influenced by experimental evidence of a TTX-insensitive inward current that does not inactivate, Plant and Kim included an inward current with constant conductance. The other components of the earlier model survived with minor modifications, resulting in the following current balance equation: \[ C_m \frac{dV}{dt} = - \bar{g}_I \cdot x^3_I \cdot y_I \cdot (V-V_I) - \bar{g}_T \cdot (V-V_I) - \bar{g}_K \cdot x^4_K \cdot (V-V_K) - \bar{g}_A \cdot x_A \cdot y_A \cdot (V-V_K) - \bar{g}_P \cdot x_P \cdot (V-V_K) - \bar{g}_L \cdot (V-V_L) + I_P. \]

Plant, 1978

In 1978, Plant substantially overhauled the current balance equation, and introduced the dependence on intracellular calcium concentration: \[ C_m \frac{dV}{dt} = - \bar{g}_I \cdot x^3_I \cdot y_I \cdot (V-V_I) - \bar{g}_T \cdot x_T \cdot (V-V_I) - \bar{g}_K \cdot x^4_K \cdot y_K \cdot (V-V_K) - \bar{g}_A \cdot x^3_A \cdot y_A \cdot (V-V_K) - \bar{g}_P \cdot \frac{ Ca }{ 0.5 + Ca } \cdot (V-V_K) - \bar{g}_L \cdot (V-V_L). \] Here, the action potentials still are generated by Hodgkin-Huxley-like currents (the first and third ion currents in the above equation), but the delayed potassium current now includes inactivation. Further, the dynamics of the (in)activation variables for this current depend on both voltage and intracellular calcium concentration. The slow, periodic oscillation in the membrane potential in the presence of TTX is governed by the interaction of the TTX-resistant inward current (second ion current in the above equation), a fast potassium current (fourth current), and a pacemaker potassium current (fifth current). The TTX-insensitive inward current is endowed with slow voltage-dependent activation properties. The fast potassium current survived from previous models, albeit in slightly modified form. The pacemaker potassium current is activated by the buildup of intracellular calcium ions.

The Plant (1981) model as presented in the section above is a simplification of this last model. In particular, the delayed potassium current is restored to the classical Hodgkin-Huxley form, and the fast potassium current is eliminated.

Stable Oscillations

In the original papers by Plant and Kim (1975, 1976) and Plant (1978), the focus of the analytical treatment of the equations was not so much to discover the mechanism underlying bursting as it was to prove the existence of a stable oscillation of the membrane potential in the presence of TTX. To achieve this, Plant and Kim determined conditions under which the model, with the TTX-sensitive current eliminated, would exhibit a slow oscillation. The approach was to demonstrate that the solution of the resulting system asymptotically approached the solution of a second-order system, the behavior of which was then studied in the phase plane. This approach was similar to that used by Krinsky and Kokoz (1973) in a study of the Hodgkin-Huxley model, and relied on a theorem of Tikhonov (1950). It turns out that the reduced two-dimensional system is analogous to a van der Pol oscillator or a relaxation oscillator, the behaviour of which was well known at that time.

Anecdote

The question of finding stable oscillations was solved due to a fortuitous coincidence. At that time, mathematicians in the former Soviet Union were widely regarded as leaders in the field of nonlinear oscillations, so Plant set about to pursue the Soviet literature for clues on how to proceed. One of the first journals he picked up was a copy of Biophysics volume 18, which had just been translated into English. In that volume was the article by Krinsky and Kokoz (1973) that discussed the application of a theorem of Tikhonov to problems in modeling excitable nerve membranes. The Tikhonov (1950) article was not available in English, but Plant succeeded in translating enough of it to make sense, and it turned out to be the perfect solution to the problem of finding conditions that ensured an oscillatory solution to nerve membrane equations.

References

• Bertram R., Butte M.J., Kiemel T., and Sherman A. (1995) Topological and phenomenological classification of bursting oscillations. Bull. Math. Biol. 57:413-439.
• Connor J.A. and Stevens C.F. (1971) Prediction of repetitive firing behavior from voltage clamp data on an isolated neurone soma. J. Physiol. 213:31-53.
• de Vries G. (1998) Multiple bifurcations in a polynomial model of bursting electrical activity. J. Nonlinear Science 8:281-316.
• Frazier, W.T., Kandel, E.R., Kuppermann, J., Waziri, R., and Coggeshall, R.E. (1967) Morphological and functional properties of identified neurons in the abdominal ganglion of Aplysia californica. J. Neurophysiol. 30:1288.
• Golubitsky M., Josic K., and Kaper T.J., (2001) An unfolding theory approach to bursting in fast-slow systems. In: Broer H.W., Krauskopf B., and Vegter G. (eds.) Global Analysis of Dynamical Systems: Festschrift dedicated to Floris Takens on the occasion of his 60th birthday, pp. 277-308, Institute of Physics, Bristol.
• Izhikevich, E.M. (2000) Neural excitability, spiking and bursting. Int. J. Bifurcation and Chaos 10:1171-1266.
• Junge, D. and Stevens, C.L. (1973) Cyclic variation of potassium conductance in the burst generating neurone in Aplysia. J. Physiol. 235:155-181.
• Krinsky V.I. and Kokoz, Y.M. (1973) Analysis of equations of excitable membranes. Biophysics 18:533-539.
• Mathieu P.A. and Roberge F.A. (1971) Characteristics of pacemaker oscillations in Aplysia neurons. Can. J. Physiol. Pharmacol. 49:787-795.
• Plant R.E. (1978) The effects of calcium++ on bursting neurons. Biophys. J. 21:217-237.
• Plant R.E. (1981) Bifurcation and resonance in a model for bursting nerve cells. J. Math. Biol. 11:15-32.
• Plant R.E. and Kim M. (1975) On the mechanism underlying bursting in the Aplysia abdominal ganglion R15 cell. Math. Biosci. 26:357-375.
• Plant R.E. and Kim M. (1976) Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations. Biophys. J. 16:227-244.
• Rinzel J. (1987) A formal classification of bursting mechanisms in excitable systems. In: Teramoto E. and Yamaguti M. (eds.) Mathematical Topics in Population Biology, Morphogenesis, and Neurosciences (Lecture Notes in Biomathematics 71:267-281), Springer, Berlin.
• Rinzel J. and Lee Y.S. (1986) On different mechanisms for membrane potential bursting. In: Othmer, H.G. (ed.) Nonlinear Oscillations in Biology and Chemistry (Lecture Notes in Biomathematics 66:19-33), Springer, Berlin.
• Rinzel J. and Lee Y.S. (1987) Dissection of a model for neuronal parabolic bursting. J. Math. Biol. 25:653-675.
• Strumwasser, F. (1971) The cellular basis of behavior in Aplysia. J. Psychiatr. Res. 8:237-257.
• Tikhonov A.N. (1950) On systems of differential equations containing parameters (in Russian). Mater. Sb. 27:147-156.

Internal references

• John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
• Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
• Frances K. Skinner (2006) Conductance-based models. Scholarpedia, 1(11):1408.
• James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
• Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
• Takashi Kanamaru (2007) Van der Pol oscillator. Scholarpedia, 2(1):2202.

See also

Aplysia R15 neuron Bursting, Bifurcations, Dynamical Systems, Equilibrium, Hodgkin-Huxley Model, Neural Oscillators, Periodic Orbit, Routes into Bursting, Stability