Aggregate input-output models of neuronal populations

Abstract

An extraordinary amount of electrophysiological data has been collected from various brain nuclei to help us understand how neural activity in one region influences another region. In this paper, we exploit the point process modeling (PPM) framework and describe a method for constructing aggregate input-output (IO) stochastic models that predict spiking activity of a population of neurons in the "output" region as a function of the spiking activity of a population of neurons in the "input" region. We first build PPMs of each output neuron as a function of all input neurons, and then cluster the output neurons using the model parameters. Output neurons that lie within the same cluster have the same functional dependence on the input neurons. We first applied our method to simulated data, and successfully uncovered the predetermined relationship between the two regions. We then applied our method to experimental data to understand the input-output relationship between motor cortical neurons and 1) somatosensory and 2) premotor cortical neurons during a behavioral task. Our aggregate IO models highlighted interesting physiological dependences including relative effects of inhibition/excitation from input neurons and extrinsic factors on output neurons.

Fig. 1

A) Original system of inputs u_i and outputs y_j , i ∈ 1, … , n, j ∈ 1, … , m. B) Reduced system of inputs u_i and outputs y_k , i ∈ 1, … , n, k ∈ 1, … , K, K ≤ m. C) and D) show a transfer matrix representation of these IO systems, with n = 3, m = 5, K = 2.

Fig. 2

A) A schematic of the functions with which input neurons have an effect on output neurons. Note that y1, … ,y30 are simultaneously affected by u1 and u2, whereas y31, … ,y45 are only affected by u3. B) The four functions denoting the propensity with which the output neuron spikes at time t given that input neuron spiked t – l ms ago.|

Fig. 3

Clustering Analysis. A) Schematic of individual IO model parameters. B) 1st and 2nd principal components of the parameter vectors estimated. Note that the principal components visually separate out Θ₁, … , Θ₁₅, … , Θ₁₆, … , Θ₃₀, and Θ₃₁, … Θ₄₅. C) Schematic of aggregate IO models after model reduction via K-means clustering.

Fig. 4

Aggregate IO Model Parameters. A) exp (${\overline{ϴ}}_1$), representing a model of the output neurons y1, …, y15. B) exp (${\overline{ϴ}}_2$), representing a model of the output neurons y16, …, y30. C) exp (${\overline{ϴ}}_3$), representing a model of the output neurons y31, …, y45. The 95% confidence bounds for the model parameters are depicted in grey around the parameters. The relevant exp (${\tilde{\beta }}_i$) are shown in dashed lines.

Fig. 5

Simultaneous neural activity was recorded from a non-human primate as it reached towards and manipulated four different objects in space.

Fig. 6

Aggregate IO Model Parameters. A) exp (${\overline{ϴ}}_1$), representing a model of the output neurons n1, … , n12. B) exp (${\overline{ϴ}}_2$), representing a model of the output neuron n13. The 95% confidence bounds for the model parameters are depicted in grey around the parameters. The crosses refer to those parameters which are significantly different from 1.

Fig. 7

Aggregate IO Model Parameters. A) exp (${\overline{ϴ}}_1$), representing a model of the output neurons n1, … , n12. B) exp (${\overline{ϴ}}_2$), representing a model of the output neuron n13. C) exp (${\overline{ϴ}}_3$), representing a model of the output neurons n14, n15. D) exp (${\overline{ϴ}}_4$), representing a model of the output neurons n16, … , n21. The 95% confidence bounds for the model parameters are depicted in grey around the parameters. The parameters that are significantly above or below 0 are depicted by crosses. Only the inputs with at least one significant parameter are shown.

Fig. 8

A) , B) , C) , D) The average firing rate at the PMv channels which are shown to have a significant effect (with a confidence of < 95%) on the respective clusters of M1 neurons k = 1, … , 4, during the four different tasks, on the left (color version available online). λ_k k = 1, … , 4 as estimated on all the trials performed during each task, ± one standard deviation in dashed lines, on the right.