Abstract
We discuss methods for optimally inferring the synaptic inputs to an electrotonically compact neuron, given intracellular voltage-clamp or current-clamp recordings from the postsynaptic cell. These methods are based on sequential Monte Carlo techniques (“particle filtering”). We demonstrate, on model data, that these methods can recover the time course of excitatory and inhibitory synaptic inputs accurately on a single trial. Depending on the observation noise level, no averaging over multiple trials may be required. However, excitatory inputs are consistently inferred more accurately than inhibitory inputs at physiological resting potentials, due to the stronger driving force associated with excitatory conductances. Once these synaptic input time courses are recovered, it becomes possible to fit (via tractable convex optimization techniques) models describing the relationship between the sensory stimulus and the observed synaptic input. We develop both parametric and nonparametric expectation–maximization (EM) algorithms that consist of alternating iterations between these synaptic recovery and model estimation steps. We employ a fast, robust convex optimization-based method to effectively initialize the filter; these fast methods may be of independent interest. The proposed methods could be applied to better understand the balance between excitation and inhibition in sensory processing in vivo.
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Notes
We assume V t is constant, or at least changes slowly enough that we may ignore capacitative effects, though these may potentially be included in the model as well.
This time constant τ i will typically be quite small. Mathematically speaking, it prevents any discontinuous jumps in the observed current, while physically, it may represent the lumped dynamics of the electrode and any non-space-clamped (electrotonically-distant) segments of the neuron.
In the case of Monte Carlo methods for computing the expectations needed in the EM algorithm, as in the particle filter employed here, the likelihood is no longer guaranteed to increase, due to random Monte Carlo error. However, given a sufficient number of samples (particles), the algorithm will still converge properly to a steady state, where the parameters “wobble” randomly around the location of the local likelihood maximum.
In the case of noisy or incomplete observations of the voltage V(t), we need to compute three additional sufficient statistics, \(E(V(t) | \theta^{i-1}, V^{\rm obs}_{0:T})\), \(E(V(t)^2 | \theta^{i-1}, V^{\rm obs}_{0:T})\), and \(E(V_{t-dt} V(t) | \theta^{i-1}, V^{\rm obs}_{0:T})\). These may be similarly estimated from the output of the particle filter, specifically Eq. (9). Finally, as noted in Huys et al. (2006) and Huys and Paninski (2009), it is possible to estimate the additional model parameters, (g l ,V l ,V E ,V I ), via straightforward quadratic programming methods, once the sufficient statistics are in hand. However, if g I and g E have free offset terms it is not possible to uniquely specify the leak parameters, (g l ,V l ), unless observations are made at a wide range of voltages. If only a single voltage is observed, then there are more free parameters than data points and the model is not uniquely identifiable.
It is important to make a note about the errorbars computed here. These are estimates of the posterior standard deviation \(Var(g_t | V^{\rm obs}_{0:T}, \hat \theta)^{1/2}\), where we have conditioned on our estimate of the parameter θ. Clearly, this will be an underestimate of our true posterior uncertainty, which should also incorporate our uncertainty about \(\hat \theta\). It is possible to employ Markov chain Monte Carlo methods to incorporate this additional uncertainty about \(\hat \theta\) (Gelman et al. 2003), but we have not yet pursued this direction.
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Acknowledgements
LP is supported by an NSF CAREER award, a McKnight Scholar award, and an Alfred P. Sloan Research Fellowship. We thank Y. Ahmadian, Q. Huys, J. Vogelstein, and P. Jercog for many helpful discussions and critical comments. We would like to thank N.B. Sawtell for providing us with the electric fish recordings. A simplified version of the fast optimization-based filter discussed in Section 2.5 was described briefly in the review article (Paninski et al. 2010).
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Paninski, L., Vidne, M., DePasquale, B. et al. Inferring synaptic inputs given a noisy voltage trace via sequential Monte Carlo methods. J Comput Neurosci 33, 1–19 (2012). https://doi.org/10.1007/s10827-011-0371-7
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DOI: https://doi.org/10.1007/s10827-011-0371-7