a geodesic method for spike train distances
DESCRIPTION
A Geodesic Method for Spike Train Distances. Neko Fisher Nathan VanderKraats. Neuron: The Device. Input: dendrites Output: axon Dendrite/axon connection = synapse. http://training.seer.cancer.gov/module_bbt/unit02_sec04_b_cells.html. Synapse Dendrites (Input) Cell Body Axon (Output). - PowerPoint PPT PresentationTRANSCRIPT
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A Geodesic Method for Spike Train Distances
Neko Fisher
Nathan VanderKraats
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Neuron: The Device
• Input: dendrites• Output: axon• Dendrite/axon connection =
synapse
http://training.seer.cancer.gov/module_bbt/unit02_sec04_b_cells.html
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InputOutput
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
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Systems of Spiking Neurons
• Spike effect decays over time– Bounded window
• Discretization
Neuron 1:Neuron 2:
Neuron 3:
time
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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
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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
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Phase Space Overview
• Degenerate States– Quiescent State– Seizure State
Quiescence
Seizure
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Phase Space Overview
• Stable States– Zones of attraction
Quiescence
SeizureStable States
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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
SeizureStable States
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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
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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)
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Spike Count
n = 0 n = 71
n = 16 n = 17
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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
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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
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Delete
Victor/Aranov: Delete/Insert
Insert
Cost: 1
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Shift within Neurons
Victor/Aranov: Shifting spikes within neurons
Cost: q|Δt|
Δt
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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)
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Victor/Aranov: Shifting spikes between neurons
Shift Between NeuronsCost: k
Not biologically correct
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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)
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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
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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
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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
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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
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NED Equation
• Basically, take minimum of all possible matchups:
…-1 -1 -1 2 5 7 9 15 20 20 20 …
…-1 -1 -1 5 9 12 20 20 20 20 20 …
-1 -1 -1 2 5 7 9 15 20 20 20
5 9 12 20 20 20 20 20 20 20 20
-1 5 9 12 20 20 20 20 20 20 20…
-1 -1 -1 -1 -1 -1 -1 -1 5 9 12
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NED Equation
• Given 2 spike trains (points) x, y, with n neurons, window size wLet xi denote the ith neuron of x
Let S(xi) denote the number of spikes in xi
Let f(xi,p,q) = (-1)p.xi.(w)q or the concatenation of p spikes at time -1 to the beginning of xi and the concatenation of q spikes at time w to the end
Let fk(.) denote the kth spike time, in order, of the above
2kk
1
)S()S(
0
))S(),S(,(f))S()S(,,(fmink
iiiiii
n
i
yx
j
xxyjyxjxii
]||[NED yx
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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 recursivelyhttp://www.enm.bris.ac.uk/staff/hinke/fourD/pix/nx1x2p2.gif
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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[xi(t-1) || x(t)] = 0 i
http://www.enm.bris.ac.uk/staff/hinke/fourD/pix/nx1x2p2.gif
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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
xy
1,000,000
x y
385,000615,000
x y
320,000295,000 170,000 215,000
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Geodesic Initialization
x y
320,000295,000 170,000 215,000
• 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 x1
– Continue until NED[x||x1] is roughly half NED[x||y].
– Repeat until all intervals are sufficiently small.– Guarantees smooth transitions from one point to next
x1
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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
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Testing
• K-means clustering– Sample points in
different attractors• Seizure versus stable
states• Rate-differentiable
stable states
– Sample points from same attractor
• Other ideas?
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The End
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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
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Dynamical Systems Overview
• Periodic attractor– (aka Limit Cycle)
• Online example (Univ of Delaware)– http://gorilla.us.udel.edu/plotapplet/examples/
LimitCycle/sample.html
Return
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Dynamical Systems Overview
• Chaotic Attractor
F(x) = 2x if 0 ≤ x ≤ ½ 2(1-x) if ½ ≤ x ≤ 1 ½x if x ≥ 1 -½x if x ≤ 0
x=0 x=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=0.8x=0.4
F(x)=0.8
Return
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Example of stable spike train (1000 neurons for 800 ms)