We will culminate our discussion of geometric- and linear stability analysis with bifurcations, which occur when the system's behavior qualitatively and fundamentally changes depending on a parameter. Bifurcation is a ubiquitous aspect of nonlinear dynamics that describes many different cell behaviors that we will study this semester. For example, as current is applied to a neuron sitting at its resting potential, the neuron slowly depolarizes (voltage increases toward zero) and ultimately passes its spike threshold and begins to discharge action potentials. Thus, the cell as a "system" has undergone a bifurcation, i.e., its behavior has fundamentally and qualitatively changed due to a regulatable parameter, namely applied current. This is the main bifurcation we will study within the next two weeks. But for today we will study some simple canonical examples of bifurcations in 1D.

All excitable cells including neurons, myocytes, pancreatic beta cells, (and others) are examined analogously to electrical circuits by applying the same physical principles used to understand batteries, capacitors, resistors, etc., which you learned about in general physics. But what makes biophysics so interesting, as you will see, is that the resistors in real cells have voltage-dependent properties, with unique and disparate kinetic profiles. Unlike the highly engineered resistors you can buy at Radioshack that do follow Ohm's Law, biophysical resistors are nonlinear, which transforms excitable cells into interesting nonlinear dynamical systems. The word nonlinear in this case refers to the electrical conductance of the resistors, which do not follow the linear relationship between current and voltage set out by Ohm's law, i.e., V=IR. To understand how excitable cells behave we will have to first study the properties of their linear analog, the simple RC circuit, which forms the basis for the parallel conductance model of cell membranes. Second, we will begin to add new resistors in parallel that reflect actual ion channels in real cells that have their own characteristic voltage-dependence and kinetics. We will apply state-space analysis, bifurcation theory, and other forms of graphical and analytic techniques to study the rich repetoire of behaviors that are possible in real cells equipped with ion channels (nonlinear), which far outstrip what RC circuits (linear) can do.

Qualitative changes in the dynamics of a system result from critical dependence on parameters. Parameters influence the structure of the vector field, and thus the number of fixed points and their stability. We incorporate a parameter into our general definition of a dynamical system by noting x′ = f(x;r), where r is the parameter. The parameter values where qualitative changes in the dynamics occur are called bifurcation points and are denoted by r_c.

Saddle-node bifurcation
The basic mechanism for the simultaneous creation and destruction of fixed points; sometimes refered to as a blue sky bifurcation because fixed points may surprisingly appear 'out of the blue sky", but also called a fold bifurcation because of the shape of the bifurcation diagram. The prototypical first-order system with a saddle-node bifurcation takes the form: x′ = r - x².

Transcritical bifurcation
Some systems in the natural and physical world require that a fixed point must exist for all values of a parameter, and can never be created or destroyed in a bifurcation. However, the stability of the fixed point in question can change, as occurs in the transcritical bifurcation. The prototypical first-order system with a transcritical bifurcation take thes form: x′ = rx - x².

Pitchfork bifurcation - supercritical
The pitchfork bifurcation applies to physical systems that have a symmetry because here fixed points appear and disappear in pairs. In the supercritical form of the pitchfork bifurcation the cubic term is restorative, i.e., tends to make the solution x(t) head back toward x=0. The symmetric fixed points that appear for r>0 are stable. The prototypical first-order system with a supercritical pitchfork bifurcation takes the form: x′ = rx - x³.

Pitchfork bifurcation - subcritical
In the subcritical form of the pitchfork bifurcation, the cubic term is destabilizing in the sense that it tends to make x(t) head away from x=0. The symmetric fixed points exist for all r<0 and are unstable. the prototypical first-order system with a subcritical pitchfork bifurcation takes the form: x′ = rx + x³.

Ohm's Law
Current and voltage are two key state variables in electrical systems. Ohm's law defines the resistance (R) of electrical material as the ratio of voltage to current, R=V/I, or conveniently expressed as V=IR. Although it is called a law, Ohm's law is a constitutive relation, which is based on empirical measurements. We define conductance (g) as the reciprocal of resistance, g=I/V. Conductance is convenient to work with, since g can be read as the slope of the IV curve, which we can measure in the lab and will define shortly.

Membrane potential
Cell voltage (V_M) is defined as V_IN - V_OUT. This means that a positive voltage change accompanies (by definition) movement from OUT to IN across the membrane, and this represents electromotive force to drive positive membrane current (by definition) in the opposite direction from IN to OUT (Note: the second definition is that postive membrane current flows from IN to OUT). This makes sense intuitively because currents flow down the voltage gradient absent any other forces acting on the ions (more about that later).

Capacitance
Membrane is bilipid that does not conduct, but rather stores and separates electric change in the form of biological ions. This property means that membrane acts as a capacitor and thus plays the role of storing charges and maintaining cell voltage, and also influences the kinetics of voltage change in excitable cells. An example of a constitutive relation, capacitance (C) is defined as the ratio of charge (Q) and voltage (V), according to C=Q/V, or frequently arranged as Q=CV. By definition, electric current (I) is the change in charge with respect to time dQ/dt. Therefore, capacity current I_C can be obtained by differentiating the capacitor law to give dQ/dt = C(dV/dt) , or I_C = CV′.

Electrical behavior of RC circuits
The lipid bilayer of the cell membrane makes up the cell capacitance and is represented in an equivalent circuit as a single capacitor. We are ignoring spacial differences of voltage for now, assuming all points on the cell interior experience the same voltage; the cell is thus isopotential. Currents can flow across the capacitor as charges accumulate on one side and charges depart from the other side; the current is I_C, as defined above. The membrane is perforated by ion channels that can pass current and lumped together form an equivalent resistor that is arranged in parallel with the cell capacitance. For now, we assume that the lumped ion channels obey Ohm's Law, thus resistive current through the membrane I_M = V_M/R_M. Kirchoff's current law (another example of a conservation law) states that the currents entering and exiting a node equal zero. In our simple model we also define an applied current (I_a) that flows from OUT to IN (Note: I_app is defined in the opposite direction from I_M and I_C because it is typically introduced by the recording electrode and is not intrinsic to the cell).

To describe the behavior of an RC circut we start with Kirchoff's current law and solve it for the very general scenario of a step current (I_app) applied from time t₀ to t:

I_C + I_M - I_app = 0

C_MV′ + V_M/R_M - I_app = 0

∫dV/(I_appR_M - V_M) = ∫dt/R_MC_M

We integrate the left from V₀ to V, and the right from t₀ to t

From the table of integrals we note that ∫dx/(ax+b) = (1/a) LN|ax+b|

-[LN(I_appR_M - V) - LN(I_aR_M - V₀)] = (t - t₀)/R_MC_M

LN[(I_appR_M - V)/(I_appR_M - V₀)] = -(t - t₀)/R_MC_M

(I_appR_M - V) = (I_appR_M - V₀) EXP(-(t - t₀)/R_MC_M)

V(t) = I_appR_M - (I_appR_M - V₀) EXP(-(t - t₀)/R_MC_M)

Now make some definitions: V_SS = I_appR_M and τ = R_MC_M

Giving the general expression for charging/discharging RC circuits,

V(t) = V_SS - (V_SS - V₀) EXP(-(t - t₀)/τ)

This can equivalently be expressed: V(t) = V_SS + (V₀ - V_SS) EXP(-(t - t₀)/τ)

This general expression can be used for both charging and discharging of the RC circuit. V_SS is approached with an exponential time course that depends on τ, often called the cell's time constant. τ is a frequently measured property. A more intuitive graphical analysis is done (as before) by plotting the vector field V′ versus V_M, and by analyzing the critical points where V′ = 0. This analysis also shows that the steady state point V_SS is stable (an attractor).