Let stimuli I₁ and I₂ be any stimuli. Let u₁ and u₂ be the respective responses in some system. And lastly, let c be any constant. A system is linear if the response u elicted by the stimuli I₁+I₂ is

u = u₁+u₂

and the response due to the stimulus cI₁ (c times the stimulus I₁) is

u = c u₁

We will focus on the former ("additive") property.

The following MPEG file illustrates the dynamics of a capacitor-inductor-resistor circuit whose power supply generates a current I(t), t=time. The measured response is the voltage u(t). The top panel are the voltages; the bottom panel are the stimuli (color coded to associated responses). Note: u = u₁+u₂

Any system which is not linear.

The following MPEG file illustrates nonlinearity in a model of cell electrical activity. Here u(t) is the transmembrane electrical potential at time t and the stimulus I(t) is an applied current. Note: it is not true that u = u₁+u₂

(Fitz-Hugh Nagumo model).

Excitability is not a mathematical term. Roughly speaking, a system is excitable if it exhibits a threshold phenomenon with respect to a stimulus ( example ). Nerve cells are classic examples of excitable systems though there are chemical ones as well. If a nerve is stimulated with a small brief stimulus (current) the response (transmembrane potential) is small and resets to rest quickly. If the brief stimuli is of a sufficient magnitude (superthreshold) a large response (called an action potential) is ellicted. Typically in excitable systems, even larger super threshold responses do not ellict an even larger response.

The following MPEG file illustrates excitability of the barnacle muscle fiber's transmembrane potential u(t) in response to an applied current I(t).

(Morris-Lecar model)

Two common qualitative behaviors in a measured reponse are stable equilibria (or fixed points) and oscillations (or periodic solutions).

If for any initial value (condition) u(0) the response u(t) approaches a constant value U as time increases then U is said to be a (globally) stable equilibrium. Similarly , if u(t) approaches a periodic oscillation (orbit) U(t) then U(t) is said to be a (globally) stable periodic orbit of the system.

The following MPEG illustrates stable equilibria and periodic solutions in an activator-inhibitor system with two chemical species. The concentrations are u(t) and v(t). The stimulus is I(t)=0. The top panel shows an approach of u(t) to a stable periodic solution for different initial concentrations. The bottom panel shows an approach to a stable equilibria.

(Model from "Mathematical Biology", J. Murray).

Notice that in the periodic case, the resulting periodic solutions are out of phase but otherwise identical. To clearly illustrate this, it is often useful to graph the concentrations against each other. Here there are two chemicals of concentration u(t) and v(t) at time t, respectivley. The following MPEG file gives a plot of u(t) versus v(t) in the periodic case. The periodic "orbit" or "attractor" is clearly evident in the first of these movies. Here is an analogous movie for the equilibria case: MPEG

In the preceding example, the oscillation of the voltage would have continued were there no hyperpolarizing (last downward) stimulus. In some systems, a stimulus will cause the system to move from one stable configuration to what appears to be another stable configuration but that after long enough time the system returns to its initial state (or close to it). This is the idea behind adaptation.

Adaptation is difficult to define precisely. It is not a mathematical term (like excitability). Some people define adaptation as the return to an initial state when the stimulus is stepped up to a constant value.

The following MPEG file illustrates adaptation in the membrane potential u(t) given a stimulus (current) which is initially zero and then stepped up to a constant value. The constant stimulus initiates oscillations whose frequency decreases slowly and then disappear altogether. After some time, the system returns to a rest state (equilibrium) near the initial rest state.

The model mimics the membrane potential of an individual pancreatic beta cell at subthreshold glucose levels with an applied current stimulus.

If a system is bistable a slow change in the stimulus can cause transitions between stable equilibria and periodic solutions. A slowly changing regulatory chemical or process can do pretty much the same thing. If this is the case then complicated "endogenous" oscillations are possible even when the stimulus I=0.

In all of the following MPEG files membrane potential u(t) and concentration c(t) of a slow regulatory chemical are graphed. In all examples, u(t) is a "fast" variable and c(t) is a "slow" variable. Fastness and slowness is a relative concept. In the examples, u(t) changes alot in a short time (spikes) even though the concentration c(t) does not change appreciably. Thus it is more the relative time rates of change of u(t) or c(t) which make them fast or slow.

Bursting This oscillation is observed in many mouse pancreatic beta cells at intermediate glucose concentrations. This particular "bursting" pattern is called square wave bursting for obvious reasons. Notice c(t) does not change much during a spike - hence is slow.

Beat In some experiments, beta cells don't burst. Instead they spike irradically - not periodically as illustrated in this example. What is interesting here is that the small amplitude oscillations are essential for maintaining this pattern. If c(t) were (experimentally) maintained at its average value these oscillations might not occur!

"Nearly" parabolic bursting Shows patterns similar to those observed in R15 snail neurons. Note here that the transitions between spiking and nonspiking does not seem to depend on the value of u(t) as in square wave bursting above! This is a "qualitative" feature.

Tapered bursting A different kind of electrical oscillation (cat pyramidal neurons?).