1. Neural Dynamics

2. The FitzHugh-Nagumo v-Nullcline (Iap=0)
W=0 v -1 1
0<W<1 v
V derivative is 0 at the hash markes
Connect the hash marks for all w to form the v-nullcline

3. Morris-Lecar Phase Plane
Red: subthreshold response, Green: impulse

4. Equilibria

5. Example of Linearization Function of 1 Variable

6. Example of Linearization Function of 2 Variables

7. Effect of Multiplying a Vector by a MatrixAx
Typical effect of
matrix-vector multiplication
Multiplication by eigenvectors
Red: 𝜆>1
Blue: 𝜆<−1

8. Tonic Spiking in Morris-Lecar Model
With enough applied current the model produces
tonic or periodic spiking

9. Recording of Tonic Spiking
Tonic response
Single response
Spiking response to injected current recorded from a
midbrain sensory neuron. From Kohashi and Carlson, 2014.

10. Examples of Depolarization Block
Two examples of depolarization block in midbrain dopamine
neurons. From Qian et al., 2014.

11. Bistability Between Steady State and Limit Cycle
An unstable limit cycle forms the separatrix

12. A Bistable System 3 1 Potential energy function
Noise can kick the system from one basin into the other

13. Morris-Lecar Bifurcation Analysis
A type-2 neural oscillator. Black: stationary branch, Red: periodic branch

14. Three Examples of Hodgkin’s Three Classes of Neurons
Hodgkin (1948) classified neurons into three types based on their response to
depolarizing current pulses
Recordings from spinal lamina 1 neurons. From Prescott, De Koninck,
and Sejnowski, PLoS Comput. Biol., 2008

15. Type 2 Dynamics with Morris-Lecar

16. A Type-1 Neural Oscillator
Oscillation period approaches infinity near the homoclinic bifurcation

17. Example of a Saddle Point in a Planar System
Unstable manifold
Stable manifold
The saddle point is at the origin in this example.

18. Example of a Saddle Point in Terms of Potential Energy
Stable manifold

19. Sub- and Super-threshold Responses

20. In Morris-Lecar the Threshold is the Middle Branch

21. Integrate-and-Fire Model Neuron
The spike is drawn by hand

22. Integrate-and-Fire with Adaptation