Separation rates for the detection of synchronization of interacting point processes in a mean field frame. Application to neuroscience

TL;DR

The paper tackles non-asymptotic detection of synchronization between interacting point processes in a mean-field neural context by analyzing permutation tests of independence. It introduces the separation quantity $\Delta_\varphi$ and derives general non-asymptotic bounds that guarantee controlled Type II error, then applies them to two models: a jittering Poisson model and a homogeneous exponential Hawkes network. For the jitter model, the separation scales with the injection rate and jitter distribution, yielding an explicit $n_{\min}$; for the Hawkes model, the analysis reveals a fundamental limit: to detect coupling, the number of trials must grow at least as the square of the hidden network size, $n \gtrsim M^2$, with detailed dependence on $\nu$, $a$, $b$, and $\delta$. Simulations confirm the sharpness of the bounds and illustrate how network size and parameter choices affect power, providing practical guidance on data requirements for neuroscience experiments. Overall, the work connects nonparametric independence testing with mean-field neural dynamics, offering concrete, non-asymptotic guidance on when synchronization detection is statistically feasible in large neural networks.

Abstract

Permutation tests have been proposed by Albert et al. (2015) to detect dependence between point processes, modeling in particular spike trains, that is the time occurrences of action potentials emitted by neurons. Our present work focuses on exhibiting a criterion on the separation rate to ensure that the Type II errors of these tests are controlled non asymptotically. This criterion is then discussed in two major models in neuroscience: the jittering Poisson model and Hawkes processes having \(M\) components interacting in a mean field frame and evolving in stationary regime. For both models, we obtain a lower bound of the size \(n\) of the sample necessary to detect the dependency between two neurons.

Figures (8)

• Figure 1: Influence of the noise and $\delta$ on the power of the test. We simulated $n=200$ trials and we replicated this simulation $N_{{\color{black}sim}}=10000$ times to estimate the power of the test. Noises (i),(ii) (iii) and (iv) are defined in Section \ref{['sec:sim_jit']}. The parameter $\delta$ varies in a regular grid between 0.05 and 0.2. The vertical line indicates the boundary of support for all noises.
• Figure 2: Influence of the injection rate and $\delta$ on the power of the test. We simulated $n=200$ trials and we replicated this simulation $N_{sim}=10000$ times to estimate the power of the test. We use noise (iv), defined in Section \ref{['sec:sim_jit']}. The parameter $\delta$ varies in a regular grid between 0.05 and 0.2. The vertical line indicates the boundary of support for all noises.
• Figure 3: Influence of the number of trial $n$ and $\delta$ on the power of the test. We simulated $n=200, 250, 300$ and $350$ trials and we replicated this simulation $N_{{\color{black}sim}}=10000$ times to estimate the power of the test. We use noise (iv), defined in Section \ref{['sec:sim_jit']}. The parameter $\delta$ varies in a regular grid between 0.05 and 0.2. The vertical line indicates the boundary of support for all noises.
• Figure 4: Influence of the number of trials $n$ and of the size of the network $M$. The number of trials $n$ varied in a regular grid between 60 and 550 and the point processes were the first two coordinates of an homogeneous network of size $M = 10, 20, 30$, with $a=3$, $b=4$ and $\delta=0.1$. We simulated $N_{{\color{black}sim}}=10000$ times the whole procedure to approximate the power of the test.
• Figure 5: The number of necessary trials grows in $M^2$. We plot with red squares $n^*$ (see Section \ref{['sec:Haw_sim']}) as a function of the size of the network $M$. The simulation setting is similar to the one of Figure \ref{['fig:Fig_Haw1']} except that $N=10000$, $a=10$, $b=20$, $\delta=0.1$. We fitted the curve by linear regression with respect to $M^2$ and superposed in black the curve $30 +0.8M^2$.

Theorems & Definitions (29)

• Lemma 2.1
• Remark 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 4.1
• Remark 4.2
• Theorem 4.3
• proof
• Remark 4.4
• Theorem 4.5