1. Basic Models in Theoretical Neuroscience Oren Shriki 2010 Differential Equations

2. 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.

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

4. 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:

5. 1 st Order Differential Equations A 1 st order differential equation has the form: For example:

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

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

8. Separable Differential Equations By substituting: We obtain:

9. Example 1

10. Example 2 Integrating the left side:

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

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

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

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

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

16. Linear Differential Equations To sum up: Where:

17. Example Solution:

18. Example (cont.)

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