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Chaos and Poincare Maps.

We emphasize that even our minimal three-variable model exhibits the complex dynamics of bursting oscillations. Moreover, because of its simplicity and the geometric viewpoint we offer, the role of each variable is clear; V and w are for fast spike generation with bistability, and Ca provides the slow modulation. Finally, the model is sufficiently robust that in certain parameter ranges, it appears to exhibit chaotic behavior. Increasing the for the calcium dependent potassium conductance, equivalent to decreasing , can switch the membrane into a repetitively firing regime. The transition between the bursting and repetitive regimes is very complicated. For example, when the burst pattern has period 4, that is every fourth burst is the same. When , behavior is aperiodic (time course not shown) whose dynamics can be described as follows. Each time that V passes a given value, here when V decreases through 0 mV, we record the concentration of calcium, as well as the value of w. For this particular model, the recorded values of w are all about . However, the value of calcium varies between 18.7 and 20.8. The solution is approximately represented by the time series of values for the calcium, Here's how we can generate the one-variable dynamic rule whose solutions approximate these time series. With initial conditions V=0 and , we specify a value for calcium, and then integrate the full differential equations until V crosses 0 again getting the next value of calcium. Thus we have a map taking a value of calcium, Ca to a new value of calcium, This map is called a Poincare Map. The entire dynamics of our burster is captured by this simple map. For this map is shown in Figure 9c. From the figure, it is evident that there is an intersection of the line y=x and That means that there is a single concentration of calcium, , to which the trajectory returns after one cycle. This corresponds to a periodic solution to the model equations. If (as is the case here) the periodic solution is unstable. This type of map is characteristic of dynamics that have chaotic behavior. That is, the successive values of calcium appear to be random and aperiodic. By reducing the three-dimensional differential equation to a simple one-dimensional iteration, we can understand the essence of the transition from constant repetitive firing to chaos as we vary a parameter, say . More details on one-dimensional maps and chaos can be found in [13], and for an application to a neuronal bursting system see [17].

next up previous
Next: Elliptic Bursters. Up: Bursting and Adaptation: Previous: Square-wave Bursters

Bard Ermentrout
Mon Jul 29 17:47:46 EDT 1996