3.2 Methods
3.2.3 Clusterless Decoding
Point process models have become highly useful in neuroscience applications in order to characterize relationships between neural spike train data and other rele-vant stimuli. Typically, these models have been used for describing spiking from individual neurons independently, but recently, marked point process models have been used to model jointly the coding properties of neural populations (Deng et al, 2015).
A marked point process is described entirely by a joint mark intensity, λ (t, ~m|Ht), defined so that
Z
M
λ (t, ~m|Ht)d~m
= lim
∆t→0
P(spike with a mark in M in (t,t + ∆t]|Ht)
∆t
(3.1)
where Htis the history of all spiking up to time t. The joint mark-intensity can be marginalized over the full mark space M to obtain the conventional conditional intensity function of the temporal point process, which is also called the ground
intensity of the marked point process: indexes the ith spike in the bin. Then, the joint likelihood of any observed set of spike counts and marks in a time bin takes the form
p(∆Nk,{~mk}|xk) ∝ tk and the mark of the ith spike during the current interval (Deng et al, 2015).
Under the assumption of independence between tetrodes, the product of this joint likelihood across the joint mark-intensity models at each of the separate tetrodes provides the full likelihood across the recorded population.
The goal of this analysis is to compare the decoding accuracy across a range of models for λ (t, ~m|Ht) using different combinations of waveform features for the marks, ~m. Let u1, . . . , uNbe a set of observed spike times, with marks ~mu1, . . . , ~muN. For any choice of marks, the form of the joint mark-intensity model will be defined
nonparametrically, using Gaussian kernel functions centered at the rats location at each spike time and mark:
λ (tˆ k, ~mik) =
1
N∑Nn=1K x(tk), x(un), bx, ~mik, ~mun, B~m
1
K∑Kj=1K x(tk), x(tj), bx (3.3) where the function K is a one dimensional Gaussian kernel in the space of the linearized position, and K is a d + 1 dimensional Gaussian kernel function in the space of the d dimensional mark and 1 dimensional linearized position. The Gaussian kernels require a chosen bandwidth for position and in each dimension of the mark space, represented in (3.3) as a scalar bxfor position, and as a matrix B~m for mark. In all models built in these analyses, bx was set to 1.5% of the linearized track (2.5 cm) and the diagonal entries of B~mwere set based on observed variability in the corresponding waveform features. This model structure has been used previously for this dataset, and exact definitions of these kernel estimators are provided in appendix 3.5.
A visualization of this encoding model for a particular tetrode is shown in Figure 3.3. A 4-dimensional waveform peak amplitude on each channel of a tetrode was used as mark. The resulting 5-dimensional model (4 mark dimensions and 1 spatial dimension) is difficult to visualize so each panel shows a different 2-dimensional projection of the estimated joint mark intensity, with the darker shades indicating higher intensity of spiking. The leftmost panels represent po-sition versus each channel amplitude, and the larger regions of dark intensity at lower peaks across position correspond to hash spiking. A few spatially
local-ized regions can be seen at higher amplitudes, indicating various place fields. The remaining panels show the intensity as functions of pairs of channel amplitudes, which do show a few separable clusters.
Figure 3.3: Encoding model over joint position and 4-dimensional mark (peak amplitude) space, shown in dimension pairs.
For any collection of spike waveform features, we construct the model in Equation (3.3) and assess the information that it provides about spatial movement through a decoding analysis. In order to decode, we combine the spike observation model in Equation (3.2) with a state model describing the rats movement on the linearized track, p(xk|xk−1), where xkis the linearized position of the rat at time tk. These transition probabilities are defined through a Markov state evolution model
p(xk|xk−1) ∼ N(xk−1, σ )
where σ = 2 cm and ∆k = 33 ms. Together, the rat movement model and the neural spiking model provide a state-space description that allows for decoding through a point process filter, which computes the probability of the rat’s position at each point in time given observed spiking data, p(xk|∆Nk, ~mk, Hk) (Brown et al, 1998;
Eden et al, 2004). We have previously shown that the solution to the point process filtering problem can be computed using the following iterative expression:
p(xk|∆N1:k, ~m1:k) ∝ p(∆Nk, ~mk|xk) ·
Z
p(xk|xk−1)p(xk−1|∆Nk−1, ~mk−1)dxk−1,
(3.4)
where ∆N1:k and ~m1:k are the sequence of observed spikes and marks from time t1 through tk (Deng et al, 2015). Starting with the density of the rat’s position from the previous time, p(xk−1|∆Nk−1, m1:k−1), we multiply by the state model, p(xk|xk−1) and integrate over xk−1. For the one dimensional linearized position variable we are decoding, we compute this integral numerically using Riemann summation. We then multiply this by the neural spiking distribution given by Equation (3.2) using the spiking model in Equation (3.3). Finally, we renormalize to obtain the full decoding probability density for the rat’s position at the current time step. The expected value of this decoding distribution is used as a point esti-mate of the rat’s position at each time, and a 95% highest posterior density (HPD) region is computed, given by HPD = {x : p(xk= x|∆N1:k, ~m1:k) > C}, where C is selected so that Pr(xk∈ HPD|∆N1:k, ~m1:k) = 0.95.
To ensure model generalizability, we carried out all of our analyses using
two-fold cross-validation. We divided the experimental data into two consecutive halves, and built the encoding model for each half, and evaluated the decoding accuracy and uncertainty on the other. We repeated the decoding analysis using each collection of waveform features, for both our clusterless spiking model and for spike sorted data.