Biological oscillators

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Consider two neurons both with period 1. Without interaction, their phase is described mathematically and graphically as follows:

\,\!\dot{\theta}_i = 1, i = 1,2

Image:Phase_oscillations_of_clocks.png

where the phase \,\!\theta_i ranges from 0 to 1 in a circular fashion; that is, it only has values in the set [0,1). The phase 1 is not included because it is equivalent to phase 0. In the graphs above, the two oscillators have different initial phases, but they have the same period of 1. This situation gave rise to a constant phase difference all throughout the two neurons’ lives.

Contents

The Mirollo-Strogatz model

The description here of the Mirollo-Strogatz model is based on Konstantinos' discussion.

In this model, each neuron sends/fires a signal to other neurons whenever it reaches a certain phase. We may implement this signal generation at the instant when the neuron goes back to phase 0. The effect of this signal is to delay or advance other neurons in their respective phases. So, if the two oscillators above interact in this manner, we get the following sample graphical result of their phase dynamics:

Image:Mirollo-Strogatz_phase_oscillators.png

The abrupt shift in the phase \,\!\theta_1 of the first oscillator occurs due to the signal coming from the second oscillator at the instant its phase \,\!\theta_2 goes back to phase 0. Due to this phase shift in the first oscillator, it reaches its firing point earlier than it would have if no phase-shifting signal occurred. In this sense, we call the phase shift as a phase advance. As implied in the sample interaction shown in the above graph, the shift caused by every firing is always an advance. But I’m not sure if the Mirollo-Strogatz model also accommodates phase delays during interactive firing, which causes the affected neuron to reach its firing point later than it would have if no phase-shifting signal occured.

Also implied in the above illustrative interaction is the immediate effect of signal to the other neuron. In the Mirollo-Strogatz model, the effect of a firing signal is not always felt immediately by the other neurons. There is some delay \,\!\tau after the signal is fired by a neuron when the phase shift in other neurons occurs. In the above illustration, \,\!\tau = 0 because the phase shift in the other neuron occurs immediately after the signal is fired.

The Beersma-Enright model[1][2]

Schematic diagram of the model
The BE model of the SCN is a conceptual model based on a simplified representation of each SCN neuron as a clock or pacer with intrinsic period consisting of two states: activity and recovery. Without the influence of other neurons, each pacer would just cycle through its activity and recovery states rhythmically. In the BE model, the neurons interact with each other through a central discriminator which sends the overall state of the system back to each neuron as shown in the schematic diagram. In the current implementation, the discriminator signal is given by the fraction of pacers that are active (i.e. in the activity state).

Effect of discriminator signal to each pacer

There are two ways in which the discriminator can affect the pacers:

  1. Phase delay: When a pacer is about to change from activity to recovery, a non-zero discriminator signal would delay the transition.
  2. Phase advance: When a pacer is near activation, a non-zero discriminator signal would push it to early activation.


Single phase oscillator with independent periodic discriminator

Consider the phase oscillation of a single pacer driven by a periodic discriminator signal. In this further simplification, the impact of the single pacer of interest on the system output is negligible. The rest of the SCN neurons are assumed to be already synchronized in a periodic manner and are not influenced by the single pacer. In this situation, we are interested on the dynamics of the single pacer and see how it is affected by changes in its own and in system parameters. The single pacer has four parameters:

  1. intrinsic period \,\!\tau_0
  2. length of activity \,\!\alpha_0
  3. maximum phase delay \,\!\varepsilon
  4. maximum phase advance \,\!\eta

while the system has two parameters:

  1. period \,\!\tau
  2. length of activity \,\!\alpha (In the succeeding simulations, this parameter is replaced by an actual parameter \,\!a which is a monotonic function of \,\!\alpha.)
Figure 1, Single phase oscillator with independent discriminator signal

As illustrated in the figure, the next transition to activity \,\!t_{i+1} can be expressed in terms of the previous transition to activity \,\!t_i and the parameters of the pacer, as follows:

\,\!t_{i+1} + \eta F\left(t_{i+1}\right) = t_i + \varepsilon F\left(t_i + \alpha_0\right) + \tau_0

The function \,\!F(t) describes the activity of the external signal and is nonnegative. If the external system is inactive, then this function is zero. Notice that the map is not explicit—it still requires finding the solution \,\!t_{i+1} in terms of \,\!t_i, \alpha_0, \tau_0, \varepsilon, and \,\!\eta. The algorithm would have a procedure for root finding for the equation of the form

\,\!x + \eta F(x) - C = 0

where \,\!C is positive according to equation (1). Suppose \,\!F(x) is a periodic function of the form shown in Figure 2.

Figure 2, Finding the next transition time

The periodic function used in Figure 2 is a vertically shifted sine function and clipped at zero so that the resulting function will only have nonnegative values. The discriminator is inactive when the function is zero and active when it is positive. Note that the parameter \,\!a is a monotonic function of \,\!\alpha such that one is 0 or 1 if the other is also 0 or 1, respectively.

\,\!F(t) = \left(\frac{\sin{\frac{2\pi(t-t_\phi)}{\tau}-1}}{2a}+1\right)H\left(\frac{\sin{\frac{2\pi(t-t_\phi)}{\tau}-1}}{2a}+1\right)

Figure 3, Finding the next transition time; multiple solutions
Figure 4, Finding the next transition time; degenerate solution
One very important question raised about this iterative map model is whether or not the dynamical system based on it is well-defined. First of all, the equation has at least one solution all the time. However, the equation may contain multiple solutions or even degenerate ones, especially when the periodic function has a slope greater than 1 (See Figures 3 and 4). This issue can be resolved by going back to the original biological model. Recall that the transition to activity occurs at the first instance of abrupt phase advance after which the oscillator would become active and would resume its linear increase with time. Therefore, the realistic solution to the mathematical problem is the smallest among multiple solutions.

Region of entrainment

References

  1. ↑ D.G.M. Beersma et al, Emergence of circadian and photoperiodic system level properties from interactions among pacemaker cells, J Bio Rhythms August 2008
  2. ↑ J.T. Enright, The timing of sleep and wakefulness, New York:Springer (1980)
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