Introduction to Mathematical Methods in Neurobiology: Dynamical Systems Oren Shriki 2009 First Order Differential Equations

Two Types of Dynamical Systems Differential equations: Describe the evolution of systems in continuous time. Difference equations / Iterated maps: Describe the evolution of systems in discrete time.

What is a Differential Equation? Any equation of the form: For example:

Order of a Differential Equation The order of a differential equation is the order of the highest derivative in the equation. A differential equation of order n has the form:

1 st Order Differential Equations A 1 st order differential equation has the form: For example:

Separable Differential Equations Separable equations have the form: For example:

Separable Differential Equations How to solve separable equations? If h(y)≠0 we can write: Integrating both sides with respect to x we obtain:

Separable Differential Equations By substituting: We obtain:

Example 1

Example 2 Integrating the left side:

Example 2 (cont.) Integrating the right side: Thus:

Linear Differential Equations The standard form of a 1 st order linear differential equation is: For example:

Linear Differential Equations General solution: Suppose we know a function v(x) such that: Multiplying the equation by v(x) we obtain:

Linear Differential Equations The condition on v(x) is: This leads to:

Linear Differential Equations The last equation will be satisfied if: This is a separable equation:

Linear Differential Equations To sum up: Where:

Example Solution:

Example (cont.)

Derivative with respect to time We denote (after Newton):

RC circuits R – Resistance (in Ohms) C – Capacitance (in Farads) I R C Current source

RC circuits I R C The dynamical equation is:

RC circuits Defining: We obtain: The general solution is:

RC circuit Response to a step current:

RC circuit Response to a step current:

Integrate-and-Fire Neuron R – Membrane Resistance (1/conductance) C – Membrane Capacitance (in Farads) I R C inside outside ELEL Threshold mechanism

Integrate-and-Fire Neuron If we define: The dynamical equation will be: To simplify, we define: Thus:

Integrate-and-Fire Neuron The threshold mechanism: –For V<θ the cell obeys its passive dynamics –For V=θ the cell fires a spike and the voltage resets to 0. After voltage reset there is a refractory period, τ R.

Integrate-and-Fire Neuron Response to a step current: IR<θ: t V

Integrate-and-Fire Neuron Response to a step current: IR>θ: V t T τRτR τRτR τRτR