An Event-Driven Approach to Modeling Excitable Cells using Hybrid Automata Mike True, SUNY at Stony Brook A Joint Work with: Emilia Entcheva Radu Grosu Scott A. Smolka Pei Ye

Excitable Cells Excitable cells are cells that respond to electrical stimuli with electrical signals known at the cellular level as Action Potentials (AP) An AP is fired by an excitable cell as an all-or-nothing response to an electrical stimulus external to the cell The sequence of events followed by an AP is for the most part independent of the magnitude of the stimulus Examples of excitable cells found in mammals include those found in cardiac tissue and neurons

Features of Action Potentials While the AP of excitable cells might vary greatly in duration and morphology, they generally exhibit the same major phases

The Full Hodgkin-Huxley Model

Hybrid Automata Models Differential equations are very expensive to compute directly in simulations Redefining our model such that all of the differential equations are linear (i.e., of the form δx = ax for some constant a) results in a significant performance improvement We propose an approach to simulation using Hybrid Automata (HA) models for cells

Hybrid Automata In our HA models, we divide AP into four different states, namely Resting, Stimulated, Upstroke, and Plateau The variables of each cell change with time in accordance with the set of differential equations associated with the cell’s current state When certain conditions are met, the HA will transition from one state to another To obtain improved performance, we add the restriction that all differential equations in all states must be linear

An HA Hodgkin-Huxley Model

Results for the HH Model

An Observation A simple simulator implementation for our HA models can be made by using Time-step Integration techniques This implementation can be improved after observing that the cell is non-responsive to external input in the Upstroke and Plateau states We can readily solve the differential equations for these states to compute how long the cell will spend in them This modification allows us to effectively ignore cells in these states for several thousands of time steps

An Event-Driven Model We associate an event with each cell to represent the next time that the cell requires processing and the type of processing required The correct ordering of events is maintained by storing events on a priority queue data structure

Types of Events Query_Neighbor: Uses time-step integration to update the cell’s variables considering the effects of neighboring cells and external stimuli (Resting and Stimulated states) Update_State: Requires a transition to the target state at the calculated time (Upstroke and Plateau states) Output_To_File: Dumps voltage values to an output file Begin/End_Stimulation: Applies/Removes stimulus currents from affected cells at the stimulus times

An Event-Driven Model Example

Priority Queue Implementation A standard data structure for implementing a priority queue would be a min-heap, where the time of the event would be the priority This data structure requires O(log 2 n) operations for insertions and removals, creating a large amount of overhead for Query_Neighbor events We propose a hybrid priority queue, where Query_Neighbor events are stored on two linked lists and all other events are stored on a min-heap

Performance Guarantees With n events on the min-heap, we can expect 2*n operations (one insertion and removal) while an event is on the heap. The total work is O(n log 2 n) operations. We can assume that each event is equally responsible for 1/n of this overhead, meaning each event is “charged” with O(log 2 n) units of work Because this work is spread throughout the entire time the event is on the min-heap, the amount of work per time-step is approximately (log 2 n)/s, where s is the number of time-steps the event spends on the min-heap

Performance Guarantees For the work required per cell per time-step in the Upstroke and Plateau states to exceed 1 unit, we would need n to be equal to 2 s Since typical values for s are in the thousands, a number of cells much greater than what can be stored in a computer’s memory system is required for this to occur In the worst case (all cells in the Resting and Stimulated states) the Time-step Integration model will outperform the Event-Driven model by a small factor related to the linked list overhead

Simulation Timing Results 400x400 Grid, 500 ms simulation (spiral wave): –Event-Driven model time: 1.7 hours –Time-step Integration model time: 9.05 hours –Speedup Factor: 5.33 Grids of various sizes, wave started in a corner: –Speedup Factor: about 1.25 –This would be considered a “worst-case scenario” for the Event- Driven model

Future Work “Putting cells to sleep” at resting potential, similar to the idea behind adaptive time-step techniques Develop tools to facilitate the rapid creation of simulators for models based upon HA specifications that use Event- Driven techniques