Important terms in Neuroscience

x-nullcline: The set of points where the vector field changes its horizontal direction is called x-nullcline, defined by equation f(x,y)=0. At such point x neither increases nor decreases because d(x)/dx = 0. X-nullcline divides the plane into 2 planes where x moves in opposite direction.

y-nullcline: The set of points where the vector field changes its vertical directions is called y-nullcline, defined by equation g(x,y)=0. At such point y neither decreases nor increases. These 2 nullclines partitions  the phase plane into 4 different regions

• X and y increase
• X decreases y increases
•  X and y decreases
• X increases and y decreases

Equilibrium point : intersection of nullclines is an equilibrium point. 

Trajectories : A solution to the Two dimensional system is a curve (x(t),y(t)) on the phase plane which is tangent to the vector field is called a trajectory or orbit.

Periodic Orbits: A trajectory that forms a closed loop s called periodic orbit, periodic trajectory or limit cycle. Periodic orbits cannot occur in one dimensional system. If the initial point is on periodic orbit the solution remains on the orbit forever and the system exhibits periodic behavior

x(t)= x(t+T)      and      y(t)= y(t+T)

The minimal value of T for which above holds is called the period or the periodic orbit.

Asymptotically stable periodic orbit: A periodic order is said to be asymptotically stable if any trajectory with the initial point sufficiently near the orbit approaches to the orbit as t-> infinity.

Limit Cycle Attractor: Asymptotically stable periodic orbits are often called limit cycle attractor, since they attract all nearby trajectories.

Limit Cycle Repeller: The unstable periodic orbit are often called a repeller, since it repels all near by trajectories

Note: There is always at least one equilibrium inside any periodic orbit on a plane.

Equilibria: Equlibria is a point where f(x,y)=0 and g(x,y)=0. Geometrically equlibria are intersection of nullclines. If the initial point (x₀,y₀) is an equilibrium, then trajectory stays at equilibrium. If the initial point is near the equilibrium, then the trajectory may converge to or diverge from the equilibrium depending on its stability.

Stability

Stability of Equilibrium : An equilibrium is stable  if any trajectory starting sufficiently close to the equilibrium remains near it for all t>=0.

Asymptotically stable equilibrium: If any trajectories starting sufficiently near the equilibrium converge to the equilibrium as t-> infinity.

Exponentially stable:  If convergence rate is exponentially fast then equilibrium is said to be exponentially stable.

Note: stability doesn’t imply asymptotically stable

Neutrally Stable:  Some trajectories neither converge nor diverge from equilibrium

Unstable Equilibrium: An equilibrium is called unstable, if it is not stable i.e if all nearby trajectories diverge from the equilibrium.