Tangent Dynamics Through Spikes, Structure, and Advanced Topics
Stub. Extending tangent-dynamics and Lyapunov-exponent analysis to hybrid spiking systems, where a perturbation's evolution must be tracked across discrete reset events as well as continuous flow; how a network's connectivity structure localizes activity or instability to particular subpopulations; and a grab-bag of selected advanced topics closing out the course.
Corresponds to the second half of the “Spiking networks and research frontiers” unit (weeks 11–13), the course’s final unit — no lecture dates posted yet; update once lectures are posted. Generalizes tangent-dynamics-chaos-and-lyapunov-exponents to hybrid spiking systems: a perturbation evolves under the smooth flow between spikes as before, but also has to be propagated correctly through each discrete reset event (where the reset map’s own Jacobian — often singular or discontinuous — enters the calculation), which complicates both the theory and the repeated-QR numerics used to compute Lyapunov spectra. Structure and localization asks how a network’s connectivity pattern (as opposed to fully random connectivity) concentrates activity, instability, or memory onto particular subsets of neurons rather than spreading it uniformly. Closes with selected advanced topics the course ends on, to be filled in once announced.
Outline (TODO — flesh out each)
- Tangent dynamics through spikes — perturbations across reset events
- Lyapunov exponents for hybrid/spiking systems
- Structure and localization in network dynamics
- Selected advanced topics (TBD)