A Geodesic Method for Spike Train Distances Neko Fisher Nathan VanderKraats

Neuron: The Device Input: dendrites Output: axon Dendrite/axon connection = synapse

Input Output Threshold Time SynapseSynapse Dendrites (Input)Dendrites (Input) Cell BodyCell Body Axon (Output)Axon (Output) Neuron: The Device Slide complements of Arunava Banerjee How is information transmitted? – Spikes – Soma’s Membrane Potential (V) – Weighted sum – Spike effect decays over time

Systems of Spiking Neurons Spike effect decays over time –Bounded window Discretization Neuron 1: Neuron 2: Neuron 3: time

Systems of Spiking Neurons Spike train windows = points in Phase Space Neuron 1: Neuron 2: Neuron 3: time Dynamical System –Each point has a well-defined point following it

Phase Space Overview Extremely high dimensionality –For systems of 1000 neurons and reasonable simulation parameters, we have up to 100,000 dimensions!! Sensitive to Initial Conditions –Chaotic attractors

Phase Space Overview Degenerate States –Quiescent State –Seizure State Quiescence Seizure

Phase Space Overview Stable States –Zones of attraction Quiescence Seizure Stable States

Phase Space Overview Problem: Given a point (spike train), how can we tell what state we’re in? –Need Distance Metric between pts in our space Quiescence Seizure Stable States

Distance Metrics Spike Count Victor/Aranov Multi-neuronal edit distance Leader in the field Our Work Neuronal Edit Distance Distance Metric using a Geodesic Path

Spike Count Count the number of spikes Can tell between quiescent, stable and seizure state spike trains Hard to differentiate between spike trains from the same state(Quiescent, Stable and Seizure)

Spike Count n = 0 n = 71 n = 16 n = 17

Edit Distance Standard for calculating distance metrics Derived from Edit Distance for genetic sequence allignment Considers number of spikes Considers temporal locality of spikes Uses standard operations on spike trains to make them equivalent Insert/delete Shift

Victor/Aranov Multi-neuronal Edit Distance Insert/Delete Cost of 1 Shifting spikes within a neuron Cost of q |Δt| Shifting spikes between neurons Cost of k

Delete Victor/Aranov: Delete/Insert Insert Cost: 1

Shift within Neurons Victor/Aranov: Shifting spikes within neurons Cost: q|Δt| ΔtΔt

Victor/Aranov: Shifting spikes within neurons D = q |Δt| q determines sensitivity to spike count or spike timing –q = 0  spike count metric –Increasing q  sensitivity to spike timing –Two spikes are comparable if within 2/q sec. q|Δt|  2 (Cost of inserting and deleting)

Victor/Aranov: Shifting spikes between neurons Shift Between Neurons Cost: k Not biologically correct

Victor/Aranov: Shifting spikes between neurons d = k k = 0  neuron producing spike is irrelevant k > 2  spikes can’t be switched between neurons (cost would be greater than inserting and deleting)

Problems with Victor/Aranov Edit Distance Allows switching spikes between neurons Insert/delete cost are constant Edit Distances are Euclidean Needs Manifold  Euclidean distance cuts through manifold  Define local Euclidean distance  Move along manifold

Our Work Respect the Phase Space –Riemannian Manifold –Geodesic for distances Better local metric –Biologically-motivated edit distance (Neuronal Edit Distance) –Modification for geodesic (Distance Metric for Geodesic Paths) Testing: simulations

NED Operations Consider operations within each neuron independently –Total Distance is sum over all neurons Which situation is better? –6 spikes moving 1 timestep each –1 spike moving 6 timesteps Reward small distances for individual spikes –Cost of shifting a spike is (Δt) 2

NED Operations Which is better? –Extra spike in the middle of the time window –Extra spike in the beginning of the time window Potential spikes just off the window edge! Insert a spike by shifting a spike from the beginning of the window –Cost: (t-(-1)) 2 Delete a spike by shifting spikes to the end of the window –Cost: (t-WINDOW_SIZE) 2

NED Equation Basically, take minimum of all possible matchups: … … … … …

NED Equation Given 2 spike trains (points) x, y, with n neurons, window size w Let x i denote the i th neuron of x Let S(x i ) denote the number of spikes in x i Let f(x i,p,q) = (-1) p.x i.(w) q or the concatenation of p spikes at time -1 to the beginning of x i and the concatenation of q spikes at time w to the end Let f k (.) denote the k th spike time, in order, of the above

Geodesic Euclidean metric only good as a local approximation Globally, need to respect the phase space System dynamics come from points advancing in time Include small time changes locally Define small Euclidean distances from any of these “close in time” points Do global distances recursively

Geodesic New Local Distance (DMGP) –Distance Metric for Geodesic Paths Given a point x(t): Next point in time should have very low distance –Compute x(t+1) –DMGP[x(t) || x(t+1] = 0 For symmetry, define previous time similarly –Compute all possibilities for x(t-1) –DMGP[x i (t-1) || x(t)] = 0  i

Geodesic Initialization Geodesic algorithm must be given starting path with a set number of timesteps How to find an initial path? Our Idea: –Trace the NED –Subdivide recursively to create a path of arbitrary length x y 1,000,000 x y 385,000615,000 x y 320, ,000170,000215,000

Geodesic Initialization x y 320, ,000170,000215,000 How to subdivide a given interval between x and y? –Randomly select individual spikes from y and move them toward x, using the minimum distance as defined by NED[x||y], to create a new point x 1 –Continue until NED[x||x 1 ] is roughly half NED[x||y]. –Repeat until all intervals are sufficiently small. –Guarantees smooth transitions from one point to next x1x1

Geodesic Algorithm Initialize For each point x  (t) along geodesic trajectory: –For some fixed NED distance , consider local neighborhood as all points x’ where {NED(x(t)||x’) <  } U {NED(x(t+1)||x’) <  } U {NED(x(t-1)||x’) <  } Repeat until total distance stops decreasing

Testing K-means clustering –Sample points in different attractors Seizure versus stable states Rate-differentiable stable states –Sample points from same attractor Other ideas?

The End

Dynamical Systems Overview Fixed-point attractor X = 4 f(x) = 2 X = 2 f(x) = 1 X = 1 f(x) = 0.5 X = 0.5 f(x) = 0.25 y = ½ x Fixed point: x = 0 Return

Dynamical Systems Overview Periodic attractor –(aka Limit Cycle) Online example (Univ of Delaware) – LimitCycle/sample.htmlhttp://gorilla.us.udel.edu/plotapplet/examples/ LimitCycle/sample.html Return

Dynamical Systems Overview Chaotic Attractor F(x) = 2xif 0 ≤ x ≤ ½ 2(1-x)if ½ ≤ x ≤ 1 ½xif x ≥ 1 -½xif x ≤ 0 x=0x=1 F(x) attracts to the interval [0,1], then settles into any of an infinite number of periodic orbits Sensitive to initial conditions –Minor change causes different orbit x=1.7 F(x)=0.85 x=0.85 F(x)=0.3 x=0.3 F(x)=0.6 x=0.6 F(x)=0.8 F(x)=0.4 x=0.8 x=0.4 F(x)=0.8 Return

Example of stable spike train (1000 neurons for 800 ms)