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Data Assimilation in a Two Neuron Network Anna Miller University of - - PowerPoint PPT Presentation
Data Assimilation in a Two Neuron Network Anna Miller University of - - PowerPoint PPT Presentation
Data Assimilation in a Two Neuron Network Anna Miller University of California San Diego December 7, 2017 Data Assimilation Objective Combine a theoretical model with experimental data to estimate physical properties of the system d x ( t
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Annealing
A(X|Y) =
M
- n=1
L
- l=1
Rm,l 2 (yl(tn) − xl(tn))2 +
M−1
- n=0
D
- d=1
Rf ,d 2 (xd(tn+1) − fd(x(tn), p))2 D is the number of state variables, L is the number of measured variables, and fd advances the state of the system in time.
◮ We start with a small Rf value, solves for the most likely state
history and parameters, then increases Rf by the formula: Rf = Rf 0αβ
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Network
Image borrowed from Homework 4:
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Model
C dV (t) dt = gNam3(t)h(t)(ENa − V (t)) + gKn4(t)(EK − V (t)) + gL(EL − V (t)) + Iinj(t) + r(t)gglu(Eex − V (t)) dx(t) dt = x∞(V (t)) − x(t) τx dr(t) dt = αrTe[V (t)](1 − r(t)) − βrr(t) x∞(V (t)) = 1 2
- 1 + tanh
V (t) − Vx0 dVx
- τx
= tx0 + tx1
- 1 − tanh2
V (t) − Vx0 dVx
- Te[V (t)]
= 1 1 + e(7−V (t))/5 where x describes the behavior of all gating variables: m,n, and h.
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Experiment Data
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Experiment Info
◮ Simulated Current Clamp ◮ Each individual neuron is NaKL with an additional synaptic
current
◮ Noise added to both the current and voltage data by pulling
from a gaussian distribution
◮ ENa = 55 mV, EK = −90 mV, Eex = −38.0 mV,
αr = 2.4 mM−1ms−1, βr = 0.56 ms−1
◮ Annealing done on 25000 time steps or 0.5 seconds of data ◮ Annealing settings: α = 1.25 and β range: 0 − 94
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Lowest Action?
We want to find the value of some function G(X): E[G(X)] =
- dXG(X)e−A(X|Y)
- dXe−A(X|Y)
This integral is difficult to do exactly, so we seek a solution, Xo, where A(X|Y) is the global minimum. This enables us to use Laplace’s approximation. Suppose f (x) has a large minima at x = xo:
- dxe−f (x) =
- dxe−f (xo)−f ′(xo)(x−xo)− 1
2 f ′′(xo)(x−xo)2
e−f (xo)
- dxe− 1
2 f ′′(xo)(x−xo)2 = e−f (xo)
- π
1 2f ′′(xo)
If f (x) has only one minima which dominates f (x), evaluating the integral with Laplace’s approximation is doable. If there are multiple, close together in magnitude, we must account for each of those when evaluating the integral!
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Action Plot: VA and VB Inputs
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Estimation Plot: VA and VB Inputs
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Results VA and VB Inputs
Parameter Bounds Estimated Actual g′
Na(nS/pF)
1, 250 123.992 120 g′
K (nS/pF)
1, 100 19.7614 20 g′
L(nS/pF)
0.01, 3 0.297296 0.3 EL (mV)
- 100, -10
- 53.0483
- 54
Vmo (mV)
- 100, -10
- 39.8379
- 40
dVm (mV−1) 0.02,1 0.0663339 0.06667 tm0 (ms) 0.01,3 0.107146 0.1 tm1 (ms) 0.01,3 0.388524 0.4 Vho (mV)
- 100, -1
- 60.0295
- 60
dVh (mV−1)
- 1,-0.02
- 0.0661491
- 0.06667
th0 (ms) 0.01,3 0.991706 1 th1 (ms) 0.01,10 7.07247 7 Vno (mV)
- 100, -1
- 55.0505
- 55
dVn (mV−1) 0.02,1 0.0332586 0.03333 tn0 (ms) 0.01,3 0.968999 1 tn1 (ms) 0.01,10 5.00038 5 Cm(pF−1) 0.02,1.0 0.039752 0.04 αe (mM−1ms−1) 1.0,5.0 2.42373 2.4 βe (ms−1) 0,2 0.558840 0.56 g′
glu (nS/pF)
0.5,2 0.994867 1.0
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Estimation/Prediction Plot: VA and VB Inputs
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Action Plot: VB Input
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Estimation Plot: VB Input
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Results VB Input
Parameter Bounds Estimated Actual g′
Na(nS/pF)
1, 250 151.711 120 g′
K (nS/pF)
1, 100 100 20 g′
L(nS/pF)
0.01, 3 0.491178 0.3 EL (mV)
- 100, -10
- 10
- 54
Vmo (mV)
- 100, -10
- 30.2863
- 40
dVm (mV−1) 0.02,1 0.0307475 0.06667 tm0 (ms) 0.01,3 0.0406433 0.1 tm1 (ms) 0.01,3 0.270122 0.4 Vho (mV)
- 100, -1
- 12.9963
- 60
dVh (mV−1)
- 1,-0.02
- 0.0760233
- 0.06667
th0 (ms) 0.01,3 3 1 th1 (ms) 0.01,10 1.70837 7 Vno (mV)
- 100, -1
- 53.8779
- 55
dVn (mV−1) 0.02,1 0.02 0.03333 tn0 (ms) 0.01,3 3 1 tn1 (ms) 0.01,10 1.70837 5 Cm(pF−1) 0.02,1.0 0.02 0.04 αe (mM−1ms−1) 1.0,5.0 2.26551 2.4 βe (ms−1) 0,2 2 0.56 g′
glu (nS/pF)
0.5,2 2 1
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Estimation/Prediction Plot: VB Input
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Conclusion
When provided with the voltages of both neurons, the data assimilation procedure is effective at estimating network parameters.
Future Experiments
◮ Add an excitatory connection from neuron B to neuron A. ◮ Increase the number of neurons in the network. ◮ Include inhibitory neurons in the network. ◮ Use a more complicated model for neurons: Sigmoid functions
instead of hyperbolic tangent, add in calcium current, etc.
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Appendix: Path Integral Formulation
We don’t know that our model is correct so we use stochastic differential equations: dx(t) dt = F(x(t), p) + η(t) where η(t) is gaussian noise. Since we don’t know the exact form
- f the noise with ηj(t) = 0 and ηi(t2)ηj(t1) = gijδ(t2 − t1) -
noise is independent at distinct times. We want to obtain an expression for the probability distribution P(x(tM)|y1:M) =
- M
- n=1
dxn−1P(yn|xn, y1:n−1)P(xn|xn−1)P(x0) where x are the state variables and y are the measured values. We can call the − log of the probabilities on the RHS of the above equation, the action P(x0:M|y1:M) = P(X|Y) ∝ exp[−A(X|Y)] where A(X|Y) is the effective action of the system, X is the state history, and Y contains all measurements
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Path Integral Formulation cont.
Assumptions
◮ Markovian Dynamics: The state at x(tn+1) = x(n + 1)
depends only on the state of the system at the previous time tn
◮ All noise is Gaussian and independent of any other noise
A(X|Y) =
M
- n=1
L
- l=1
Rm,l 2 (yl(tn) − xl(tn))2 +
M−1
- n=0
D
- d=1