SOQL 201 for Admins & Developers: Slice & Dice Your Org’s Data With Aggregate...
Glssm retina
1. Modeling the impact of common noise inputs of the
network activity of retinal ganglion cells
Vidne et al, J Comp. Neurosci. 2011
2. Overview
• Generalized Linear Model (GLM)
• Derive model, motivate use
• Fit model to neural population data
• Activity of Retinal Ganglion Cells (RGC) with common noise
• Test multiple network topologies
• Evaluate models, make inference on underlying networks
3. GLM
• Statistical models for neural encoding, decoding,
and optimal stimulus design
• Paninski, Pillow, Lewi (2006)
• Modelling becomes a likelihood problem
• Given a network, calculate the probability of encountering
a particular spike train.
• If you have the spike train, you can find the most likely
network.
4. Retina Physiology
• Input Layer
• Non-spiking Cells,
`analog’ activity
• Inner (noise) Layer
• Many cell types,
hard to record
• Output Layer
• Spiking RGCs
• Project to LGN, V1
5. GLM Continued
• Likelihood model expanded
• 5x5 pixels, 30 frames in time
• 10 Cosine basis functions
• 4 Cosine basis functions
• Relatively few parameters
6. Common Noise Input
• Auto-Regressive Gaussian Process
• Trong and Rieke (2008)
• RGCs share a common
noise … time scale of
about 4ms even in the
absence of modulations
in the visual stimuli
• Mixing Matrix injects noise into cells
• Generalized Linear State-Space Model (GLSSM)
7. Models (1 of 3)
• Common Noise, with coupling
• Fully Connected
• Captures input, noise,
and output of Retina
• Many parameters
8. Models (2 of 3)
• Pairwise
• Analyze cell pairs
• One noise shared
• Fit over 27 RGCs
• Poisson Model
9. Models (3 of 3)
• Common Noise, No coupling
• Disabled interactions
• Common noise
is weak compared
to other inputs
• Agrees with some
physiology (Trong
& Rieke 2008)
• Bernoulli Model
10. Model Parameter Estimation
• Preliminary ‘Private’ estimate
• Maximize the likelihood
• Estimate spatial noise covariance
• White noise stimulus to derive neuron covariance (APP. C)
• A rough estimate
• Maximize marginal likelihood
• (APP. A,B)
11. Methods
• Terminally anesthetized
• Planar array of 512 extracellular microelectrodes
• Digitized to 20kHz
• Parasol RGCs
• Magnocellular pathway, rods and cones (only 10% of RGCs)
• Large center-surround receptive fields
• Dynamic checkerboard (white noise) stimulus
• Mapped RFs of ON (n=104) and OFF (173) RGCs
• Near-complete mosaic of visual space
20. Potential Improvements
• Modify common noise
• Dynamic time-scales
• Noise scale depends on stimulus
• Mixing matrix is highly underdetermined
• Incorporate biological constraints (sparseness,
photoreceptor density) to uniquely determine MM
• Model the other 90% of RGCs
• Many types of RGCs other than parasol, i.e. midget
(parvocellular, often linked to a single cone) or Bistratified
21. Discussion
• Multi-Neuron firing patterns in large networks can be
accurately modelled
• Common-noise model, no direct coupling
• When fitting large networks, over fitting common
• Method is parallelizable, tractable, robust
• Common noise regularizes fitting (stochastic blur?)
• Agrees with synaptic noise in bipolar layer
• Possible functional advantage: common noise increases
accuracy with which some RGCs encode light (dithering)
22. Benefit
• A great tool for analyzing neural networks
• Tractable Bayesian framework
• Decoding analysis, evaluating topologies
• Infer the underlying structure of networks
• Can (will) be applied to
explore network dynamics
• Highly adaptable
• Shovel-ready
Editor's Notes
-The GLM model was proposed by paninski and pillow as a simple yet biologically plausible model for general neural networks.-In this framework, modelling becomes a likelihood problem: - given a network and stimuli, you can write down how probable it is to encounter a particular spike train - turning it on its head, you can ask, given this spike train, which network is most likely
Kind of like a random walk, where you have to stick around zero. The dependence on the process’s previous draws are captured by phi “Since the inner layers of the retina are composed of non-spiking neurons, and since each RGC receives inputs from many inner layer cells, restricting q to be a Gaussian process seems to be a reasonable starting point.”
-The likelihoodmaximation is a strongly nonlinear process. But Pillow and Paninski have shown that the likelihood is convex with respect to the parameter space, with a weak assumption made on the nonlinear transformation, specifically that it is log-convex. Thus, gradient descent can efficiently recover optimal model parameters-A correlation matrix that captures the autoregressive noise process is learned using the Yule-Walker method, it’s a very standard technique for learning the optimal AR parameters to explain a particular process-The optimization of the margnial likelihood is quite mysterious. GO TO NOTES AND EXPLAIN THE LAPLACIAN APPROXIMATION
Retinas were extracted from macaques before euthanasia
B – history filter of one ON and one OFF Ganglion Cells Note the refractory periodD – Note the weak coupling between neurons, compared to E, common noiseF – Cross-correlation of model to un-trained data suggests lack of over fittingG – Correlation function ??? Suggests that these cells are highly synchronous
A – circles are common noise, crosses are cross coupling. Linear relationship between distance and input magnitudes. Cross-coupling less important than common noise.B – common noise tends to account for about half of stimulus+post-spike inputs combinedCOMMEND strict delay in coupling filters (above 3.2)
Skip Fig. 4, shows correlation between common noise magnitude and RF overlap-A key aspect of their model is that they do not use the estimated common noise in cross-validation. The model is probabilistic, so when sampling noise they can not expect spike trains to be identical, but rather to preserve the statistical structure, which is what they show in Fig 5. Shows a strong effect of cross-coupling. 48 cells, real and predicted
Figure 6 takes the same information as the previous plot, but now shows the correlations between 3 neighbouring cellsSkip Fig. 7, shows how you can estimate the common noise q from the model and spike train, indicates that their method is working, the noise can be recovered.
So For figure 8, my jaw hit the floor when I saw these. You can see a clear quantitative and qualitative agreement between the model and data. I don’t think I’ve ever seen such a fit.COMMENT high correlation
COMMENT high correlation
Decoding – Given the spike train (D) and model parameters (common noise, A), is it possible to infer what the stimuli’s input was? Yes.B,C black – a true stimulus filtered by RF k.B,C blue/green – estimated input. Given the stimulus, you could then recover the cell’s RF.
So there are a bunch of models, which one is better, AKA which one is the brain usingA – MSE with cross-coupling OR common noise. Highly correlated, indicating both models to be similarly effectiveB – Cross-coupling models are less able to decode input if spike-times are altered. E.g., even if the spike trains have the same statistical structure, input can not be recovered. Thus, the common-noise model is a more robust model (every cell pair examined.B – there is a relationship between the loss of performance and correlation lag, indicating more distant neurons are more sensitive to cross-coupling, which suggests common noise to be more likely (?)
Bistratified, some RGCs are themselves photoreceptive
In a perfect world, the model would have 100 GLM units, fully coupled, with connections inferred from physiology. But this simply has too many parameters, would be a nightmare to try and implementInstead, acknowledge that there are parts of the model you don’t know exactly, but you do know their statistics. Incorporate these statistics to loosen up the model. Increase robustness.An analogy to image processing is dithering, adding noise to an image
