Introduction
The field of mathematical biology seeks to model biological processes using mathematical techniques and computer simulations to better explain, analyze, and visualize biological processes. Mathematical techniques often include the use of differential equations, which model events in continuous time, or difference equations, which model events in discrete time. Mathematical modeling of brain activity falls under the exciting new field of Computational Neuroscience. This interdisciplinary science focuses on analyzing the dynamics and physiology of neurons and neural networks through mathematical modeling. Early accomplishments in the field include the famous Hodgkin-Huxley model, which first modeled how an action potential is initiated and propagated. This revolutionary work has led to more complicated systems that model entire neural networks. Accomplishments in this field have proved crucial to understanding brain function. For the purpose of this paper, we will utilize differential equations to model the action potential phenomena, bursting, within the brain.
Cells of the nervous system act coordinately in networks to transmit nerve impulses throughout the body1. Understanding these neuronal dynamics can help characterize pathological behavior because disease states often result when these networks are functioning improperly. Normal and diseased states can be examined through methods of mathematical modeling. To characterize the pathology regarding neuronal functioning, we specifically looked at the dynamics of an action potential.
Neurons actively maintain their resting potential via transmembrane sodium-potassium pumps. At resting potential, the membrane potential is negative because the interior of the cell has a high concentration of potassium anions and the exterior has a high concentration of sodium cations2. In responseto a stimulus, transmembrane sodium channels open, allowing sodium to enter and depolarize the cell2. This depolarization triggers the closing of the sodium channels and opening of potassium channels2. Potassium leaves the cell, overshooting resting potential andmthereby hyperpolarizing the cell2. The sodium-potassium pumps reestablish resting potential and this process, known as an action potential, is able to repeat.
A simple model for the production of action potentials was created by Catherine Morris and Harold Lecar. This model considers calcium, potassium, and leak channels and their effect on cellular voltage. The Morris-Lecar model generates spiking at regular intervals when a current above some critical value is applied to the system. This simple model can be modified to generate a neuronal behavior called “bursting.” Bursting refers to intermittent periods of spiking and resting in a neuron. This behavior is of physiological importance because in certain bio-mechanisms it is required for normal functioning. We will discuss mathematical models for two types of bursting: square wave bursting and parabolic bursting.
All bursting models take the general form of equations 1 and 24:
In these general equations, x∈R are fast variables, y∈R are slow variables, and ϵ is a small positive parameter4. The fast variables (x) are responsible for generation of spiking, while the slow variables (y) regulate the spiking and resting phases5. In the models we used to generate square wave and parabolic bursting, the fast dynamics come from the Morris-Lecar equations. For square wave bursting, an additional negative feedback slow current is added. For parabolic bursting, there are two slow variables4: one for slow negative feedback and a second for slow positive feedback.
Methods
The Morris-Lecar equations involve three channels: calcium (Ca), potassium (K), and leak (L)6 (equation 3–5). The model exhibits periodic spiking for some applied current, Iapp, above a critical value. The Morris-Lecar model is of interest because, with modification, it can exhibit bursting behavior.
Square Wave Bursters
Rinzel and Ermentrout modify the Morris-Lecar equations by adding negative feedback (equation 6). This negative feedback responds to cellular firing and depolarization by decreasing current, thereby inhibiting and eventually stopping firing. When the membrane is resting, current gradually increases until firing begins again. The result of this negative feedback is intermittent cellular firing5.
IKCa (equation 6) is the calcium-dependent potassium current, gKCa (equation 6) is the maximal conductance for IKCa, and z is the gating variable (equations 6, 7). With these conditions, we obtain the balanced equation for calcium dynamics (equation 8). µ converts current into concentration flux, and ε is a small parameter responsible for keeping calcium dynamics slow. Equations 3–8 were used to generate our figures. The parameters used for our trials are included in Table 13.
Parabolic Bursters
Parabolic bursters differ from square wave bursters in that they have two slow variables, as opposed to one. The second slow variable is competing with the first: one provides negative feedback to the system, while the other provides positive feedback. These competing feedback systems correspond to a slow inward calcium current and a slow thereby initiating the outward current. The outward current causes the cessation of firing and the cycle repeats.
To simulate parabolic bursting we added a positive feedback inward current (equation 9) to the previously defined square-wave burster model (equations 3–8). In this model, there are two slow gating variables: s and z. z is as described previously. s fills an analogous role, but for the second slow variable.
Table 1: Parameter Values
Results
Using the equations and parameters described, we recreated the Morris-Lecar model, square wave, and parabolic bursting models. Figure 1 depicts a non-bursting model, achieved from the unmodified Morris-Lecar equations. We used this model as a framework for our square-wave, and parabolic bursting models.
Figure 1: Morris-Lecar model of an action potential.
Figure 2 depicts square-wave bursting. In this model, there is a period of resting following by a period of repetitive firing. This bursting behavior is due to slow calcium-potassium dynamics. As the neuron is firing, more potassium channels open and eventually hyperpolarize the cell. When the cell is no longer firing, calcium concentration levels decrease. Decreased calcium levels cause the potassium channels to close and, consequently, voltage to rise. The cycle then repeats.
Figure 2: Square-wave bursting. The red line depicts calcium concentration and the blue line depicts the voltage.
Figure 3 depicts parabolic bursting. There is a slow inward calcium current that activates cellular firing. In response to this firing, there is a slow outward potassium current that shuts the membrane down6. The biology of the potassium current is the same as described above in square-wave bursting. Parabolic bursting contrasts square-wave burstingin that it has an additional slow variable. Additionally, the repetitive firing of parabolic bursting occurs more rapidly, and with a longer duration, than in the slow wave bursting model.
Discussion
Understanding different forms of bursting can have major clinical applications, as certain pathologies are associated with defective neuronal bursting. For example, square wave bursting is observed in pancreatic b cells in the islets of Langerhans7. b cells produce insulin in response to glucose levels. When glucose concentrations rise, ATP levels rise inside the cell, stimulating the closure of K-ATP channels7. This depolarizes the membrane and activates voltage dependent calcium currents, leading to an influx of Ca2+. This calcium depolarizes the membraneenough to begin bursting and release insulin7. The membrane hyperpolarizes due to calcium activated potassium currents, and the membrane enters a silent phase. This creates an oscillatory bursting pattern8.
In another physiological application, parabolic bursting can be used to understand proper functioning of certain endocrine cells8. The parabolic bursting in gonadotropin releasing cells is essential for hormone release. This behavior is essential because gonadotropin releasing hormone (GnRH) is necessary for proper release of other hormones throughout the body8. GnRH releasing cells intrinsically generate oscillations, which are below the critical threshold to release GnRH8. The administration of estradiol causes the cell to experience parabolic bursting. While parabolically bursting, the cell reaches threshold and is able to release its hormones8.
The two models of bursting we studied were square-wave and parabolic bursting.The spikes generated in square-wave bursting decrease in frequency as the firing approaches the resting phase. This is due to the negative feedback slowly decreasing current. Parabolic bursting incorporates an additional slow current: a slow inward calcium current. We detailed a mathematical model for each type of bursting, though other possible models exist. These models can help to function as the basis for insights into various neural dynamics, and to study the differences between normal and pathological functioning in cells.<
Bibliography
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