Dynamics of Cellular Aggregates: Some Examples

The electrical responses of most cells are bistable with respect to electrical stimuli. Some chemical systems are bistable with respect to certain stimuli. Even mathematical models of speci populations can exhibit bistability (with respect to changes in the environment or food source). Most of these systems have a spatial element to them. For example, in chemical systems one may need to consider chemical gradients due to diffusion. In tissues, cells have a position x in space. A measured electrical response in an axon depends on at what position x along its length you place the measuring probe.

It is therefore of great interest to examine what spatial-temporal patterns can emerge as consequence of the coupling between bistable (or even monostable) units. In tissues, the "units" are cells (or parts of cells, i.e, spines versus dendrites versus soma versus axon). In biochemical systems, diffusion or active transport processes may couple units such as the ER, the intracellular space and the plasma membrane.

The electrical activity of cellular aggregates depends on the way that the cells are coupled. There are three important ways cells are coupled electrically:

  • Gap junctions
  • Synaptic coupling
  • extracellular diffusion

Cells coupled by gap junctions share current with other neighbouring cells. This is an example of "local" or "nearest neighbour" coupling since only the closest cells are affected. The shared current can act like a stimulus on the neighbouring cells and spatial-temporal patterns can arise in the cellular aggregate.

For cells coupled by synapses, currents are also "shared". Usually one thinks of this as a pre-synaptic axonal current either "depolarizes" or "hyperpolarizes" the post-synaptic cell. In other words the current is more monodirectional. In contrast to gap junction coupling, synaptic coupling is "nonlocal" since the coupled cells may be a considerable distance apart.

Lastly, extracellular diffusion of ions or chemicals can indirectly couple cells electrically. For example, if cell A undergoes an action potential an outward potassium current tends to increase the extracellular potassium concentration [K+] o (local to cell A). Diffusion would then tend equilibriate the resulting potassium gradient with respect to a neighbouring cell B. With [K+] ohigher near cell B, its electrical activity is then affected (depolarize??). The same sort of phenomena can occur with extracellular "second messengers" which bind to receptor sites busequently affecting electrical activity.

Here, we show a few spatial patterns which can arise when bistable systems are coupled.

Bistability and Travelling Wavefronts

In the following simulation, bistable cells are weakly coupled via gap junctions (here "weakly" means the conductance of the junctions is small). A spatially and temporally localized current (x=1,t=30) stimulates the aggregate. The vertical axis is time, the horizontal axis is cell position x, and the color is the "electrical potential" u(x,t). Note that a "wavefront" propagates through the medium. The "front" connects two stable synchronous states in the system. Initially all cells have the same potential. By t=100, all cells again have the same potential...but a different value.

Note that even though the coupling is weak, the response is huge.

Here's the associated movie MPEG . The movie shows the potential u versus x in time. The cell at the right is stimulated first and because it is electrically coupled to neighbouring cells, the effect of the stimulation propogates throughout the medium/tissue.

Excitability and Travelling Pulses

In the following simulation the Hodgkin-Huxley model for action potential propogation in the squid giant axon was used. The spatial variable x is the length along the axon. In all simulations the transmembrane potential is displayed.

The "coupling" between different locations x along the axon is due to diffusion. Locally high ionic concentrations are presumed to diffuse along the axon causing a current along the axon length. The model, however, is mathematically similar (almost identical) to a collection of cells i=1,2,3,... (each located at some position xi) coupled by gap junctions. In the following simulations either interpretation is valid.

Lastly, in all simulations the diffusivity is small!

First, we consider the effect of injecting a current pulse into the axon at x=1 and time t=10 (msec). The duration and amplitude of the localized pulse near x=1 is of sufficient magnitude to excite the membrane. What ensues is a localized pulse of electrical activity travelling with constant speed from x=1 to x=0. (How can you tell from the figure the pulse is travelling with constant speed?)

Here's the associated movie MPEG This really happens in the squid axon (as well as other axons, dendrites etc.)

A convenience of mathematical models is the ability to anticipate outcomes without performing the experiment. Suppose for instance each part of the axon (or cell located at x i) were not electricaly coupled and only the ionic channels let current cross the membrane. The coupling (or diffusivity) is then zero. With an identical stimulus, this is what happens:

Considering the excitability of the squid axon with respect to electrical stimuli and the the current pulse (shown below in (x,t)) this graph makes sense.

The point here is that without diffusion the action potential could not propogate down the axon! This is true despite the smallness of the diffusion. In fact, if diffusion were too large this phenomena would not be observed.

So, what if transmembrane currents were shut off with channel blockers and we wanted to observe the effect of the diffusive currents alone? Here's what happens:

Not only is diffusion alone not the cause of the travelling pulse, the spread of the electrical activity in the x direction is slower and diminishes in x.

Here's a phenomena in which

  • Small diffusivity is essential
  • the resulting propagation speed is faster than diffusion

This is a common behavior observed in "Reaction-Diffusion" systems where chemicals diffuse and react with other agents. Small diffusion can cause a very measureable and apparently rapid transport (facilitated diffusion). Of course, it is only the observed wave which is rapid. Spatial transport in the direction of the wave may be small. The phenomena is similar to water waves. Water does not move much in the direction of a water wave.

Other Examples of Wave Propagation

Electrical activity in thalamic neurons with inhibitory coupling (Ermentrout): shows two pulses anhililate each other.

Calcium propagation in heart myocytes (Keizer/Smith):

Calcium sparks in heart myocytes (Keizer/Smith):