Multi-scale detection of rate changes in spike trains with weak dependencies

Abstract

The statistical analysis of neuronal spike trains by models of point processes often relies on the assumption of constant process parameters. However, it is a well-known problem that the parameters of empirical spike trains can be highly variable, such as for example the firing rate. In order to test the null hypothesis of a constant rate and to estimate the change points, a Multiple Filter Test (MFT) and a corresponding algorithm (MFA) have been proposed that can be applied under the assumption of independent inter spike intervals (ISIs). As empirical spike trains often show weak dependencies in the correlation structure of ISIs, we extend the MFT here to point processes associated with short range dependencies. By specifically estimating serial dependencies in the test statistic, we show that the new MFT can be applied to a variety of empirical firing patterns, including positive and negative serial correlations as well as tonic and bursty firing. The new MFT is applied to a data set of empirical spike trains with serial correlations, and simulations show improved performance against methods that assume independence. In case of positive correlations, our new MFT is necessary to reduce the number of false positives, which can be highly enhanced when falsely assuming independence. For the frequent case of negative correlations, the new MFT shows an improved detection probability of change points and thus, also a higher potential of signal extraction from noisy spike trains.

Appendix: A Proofs

Here we show consistency of the estimators $\hat s^{2}$ of s ² in Eqs. (16) and (17). Recall that these were

$$\begin{array}{@{}rcl@{}} \text{global estimator:} \quad & 2h\hat\rho^{2}/\hat\mu^{3} \quad & \text{(see Lemma A.1)}\\ \text{local estimator:} \quad &\left( \frac{\hat\rho_{\text{ri}}^{2}}{\hat\mu_{\text{ri}}^{3}}+\frac{\hat\rho_{\text{le}}^{2}}{\hat\mu_{\text{le}}^{3}}\right)h \quad & \text{(see Lemma A.2)} \end{array} $$

The used estimators $\hat \rho , \hat \mu , \hat \rho _{\text {le}}, \hat \rho _{\text {ri}}, \hat \mu _{\text {le}}, \hat \mu _{\text {ri}}$ are the empirical means and estimates of ρ given in Eq. (14), derived from the whole process in the global estimator and from the local right and left windows at time t in the local estimator.

Lemma 1.1

Let {ξ _i } _i≥1 be an m-dependent process in $\mathcal P$ and $(\hat s_{nh,nt}^{2})_{t}$ the global estimator as in Eq. (16). Then it holds in (D[h, T − h], d _∥⋅∥ ) almost surely as n → ∞ that

$$(\hat s_{nh,nt}^{2} /n)_{t}\longrightarrow (2h\rho^{2}/\mu^{3})_{t}, $$

where d _∥⋅∥ denotes the supremum norm.

Proof

Note that the global estimator $\hat s$ does not depend on h and t, i.e., the formulation of $\hat s$ as a process is artificial. We show that $\hat \mu \to \mu$ a.s. and $\hat \rho _{\ell }\to \rho _{\ell }$ a.s. as n → ∞ for ℓ = 0,1,2,… where ρ ₀ = σ ². Since {ξ _i } _{i ≥ 1} is m-dependent and square-integrable, the sequence {ξ _i ξ _{i + ℓ} } _{i ≥ 1} is integrable and (m + ℓ)-dependent, thus ergodic. Then, the ergodic theorem, see e.g., Klenke (2008), states almost surely as n → ∞

$$ \frac{1}{n}\sum\limits_{i=1}^{n} \xi_{i} \!\longrightarrow\! \mathbb{E}[\xi_{1}]\,=\,\mu \quad\text{and}\quad \frac{1}{n}\sum\limits_{i=1}^{n} \xi_{i}\xi_{i+\ell}\!\longrightarrow\! \mathbb{E}[\xi_{1}\xi_{1+\ell}]. $$

(19)

Since the life times are a.s. positive and integrable, it follows N _{n T} → ∞ a.s. as n → ∞ (cmp. the proof to Lemma A.1. in Messer et al. (2014)). Thus, in Eq. (19), the value n can be exchanged with the random number of observations N _{n T} (respectively N _{n T} − (ℓ − 1)). Hence, for n → ∞, we find $\hat \mu \to \mu$ a.s. and $\hat \rho _{\ell }\to \rho _{\ell }$ a.s., so that the finite sum $\hat \rho ^{2}\to \rho ^{2}$ a.s. By construction of $\hat s^{2}$ the statement holds. □

Lemma 1.2

Let {ξ _i } _i≥1 be an m-dependent process in $\mathcal P$ and for all T>0 and h∈(0,T/2] let $((\hat s_{nh,nt})^{2})_{t}$ be the local estimator as in Eq. (17). Then it holds in (D[h,T−h],d _∥⋅∥ ) almost surely as n→∞ that $((\hat s_{nh,nt})^{2} /n)_{t}\to (2h\rho ^{2}/\mu ^{3})_{t}$.

Proof

We show the uniform a.s. convergence of $(\hat {\mu }_{\text {le}})_{t}$ and $(\hat {\mu }_{\text {ri}})_{t}$ to the constant μ in Lemma A.4, and the uniform a.s. convergence of the summands $(\hat \rho _{\text {le},\ell })_{t}$ and $(\hat \rho _{\text {ri},\ell })_{t}$ of $\hat \rho _{\text {le}}^{2}$ and $\hat \rho _{\text {ri}}^{2}$ to the constant ρ _ℓ in Lemma A.5. This implies the statement, since uniform almost sure convergence interchanges with finite sums in general and with products if the limits are constant. □

We start with a uniform a.s. result for the scaled counting process (N _t ) _{t ≥ 0}. Throughout, we use the following approach: First, we state an almost sure convergence result for the finite dimensional marginals of the processes. This essentially results from the ergodic theorem. Then, by a discretization argument, we show uniform a.s. convergence.

Lemma 1.3

Let {ξ _i } _{i ≥ 1} be a process in $\mathcal {P}$ with $\mathbb {E}[\xi _{1}]=\mu$ . Then we have in (D[h, T − h], d _∥⋅∥ ) almost surely as n → ∞ that

$$\begin{array}{@{}rcl@{}} \left( \frac{N_{nt}-N_{n(t-h)}}{nh/\mu}\right)_{t} &\longrightarrow (1)_{t}, \end{array} $$

(20)

$$\begin{array}{@{}rcl@{}} \left( \frac{N_{n(t+h)}-N_{nt}}{nh/\mu}\right)_{t} &\longrightarrow (1)_{t}. \end{array} $$

(21)

Proof

We show Eqs. (21); (20) follows analogously.

For $S_{n} := {\sum }_{i=1}^{n}\xi _{i}$ for n ≥ 1, the ergodic theorem implies S _n /n → μ a.s. for n → ∞. As we have N _t → ∞ a.s. as t → ∞, $S_{N_{t}}/N_{t}\to \mu$ a.s. as t → ∞. Now, for all t ≥ 0 we find $S_{N_{t}} \le t \le S_{N_{t} + 1}$, so that (for all t sufficiently large such that N _t ≥1)

$$\frac{S_{N_{t}}}{N_{t}} \le \frac{t}{N_{t}} \le \frac{S_{N_{t}+1}}{N_{t}+1}\frac{N_{t}+1}{N_{t}}. $$

Since the left hand side and the right hand side tend to μ almost surely we obtain N _t /t → 1/μ a.s. as t → ∞. For 0≤s<t, this implies, as n → ∞, almost surely

$$\begin{array}{@{}rcl@{}} \frac{N_{nt} - N_{ns}}{n(t-s)}& =& \frac{t}{t-s}\frac{N_{nt}}{nt} - \frac{s}{t-s}\frac{N_{ns}}{ns}\\ & \longrightarrow& \frac{t}{t-s}\frac{1}{\mu} -\frac{s}{t-s}\frac{1}{\mu} = \frac{1}{\mu}. \end{array} $$

(22)

This implies the convergence of the finite dimensional marginal of Eq. (21). The uniform convergence follows by a discretization argument analogously to the proof of Lemma A.14 in Messer et al. (2014). □

Next, we show the uniform a.s. convergence of the estimators $(\hat \mu _{\text {ri}})_{t}$, $(\hat \mu _{\text {le}})_{t}$, $(\hat \sigma _{\text {ri}}^{2})_{t}$ and $(\hat \sigma _{\text {le}}^{2})_{t}$.

Lemma 1.4

Let $\{\xi _{i}\}_{i\ge 1} \in \mathcal P$ with $\mu :=\mathbb {E}[\xi _{1}]$ . Then it holds in (D[h,T−h],d _∥⋅∥ ) almost surely as n → ∞ that

$$\left( \hat\mu_{\text{le}}\right)_{t} \longrightarrow (\mu)_{t} \qquad\text{and}\qquad (\hat\mu_{\text{ri}})_{t} \longrightarrow (\mu)_{t}. $$

Proof

Again we prove the statement only for the right window. We find $(1/n){\sum }_{i=1}^{n}\xi _{i}\to \mu$ a.s., such that $(1/N_{t}){\sum }_{i=1}^{N_{t}}\xi _{i}\to \mu$ a.s. as n → ∞. Then we conclude for all 0<s<t (the case s = 0 being similar) as n → ∞ almost surely

$$\begin{array}{@{}rcl@{}} &&\frac{1}{N_{nt}-N_{ns}} \sum\limits_{i=N_{ns}+1}^{N_{nt}} \xi_{i} \\ &&\quad\quad\quad\quad= \frac{N_{nt}}{N_{nt}-N_{ns}} \left( \frac{1}{N_{nt}}\sum\limits_{i=1}^{N_{nt}} \xi_{i} - \frac{N_{ns}}{N_{nt}}\frac{1}{N_{ns}} \sum\limits_{i=1}^{N_{ns}} \xi_{i} \right)\\ &&\quad\quad\quad\quad\longrightarrow \frac{t}{t-s}\left( \mu - \frac{s}{t} \mu\right) = \mu, \end{array} $$

(23)

making use of Lemma A.3. Thus, for every fixed t we obtain almost surely as n → ∞

$$ \hat{\mu}_{\text{ri}} =\frac{1}{N_{n(t+h)}-N_{nt}-1}\sum\limits_{i=N_{nt}+2}^{N_{n(t+h)}}\xi_{i}\longrightarrow \mu. $$

(24)

The a.s. convergence holds for finitely many t simultaneously. As above, the uniform convergence follows by a discretization argument analogously to the proof of Lemma A.15 in Messer et al. (2014). □

Now we show the uniform a.s. convergence of covariance estimators.

Lemma 1.5

Let $\{\xi _{i}\}_{i\ge 1}\in \mathcal P$ , and let $\hat \rho _{\text {le},\ell }$ and $\hat \rho _{\text {ri},\ell }$ be the local estimators of ρ _ℓ in the left and right window, see Eqs. (14), (17), for ℓ= 0, 1, 2…, where ρ ₀ =σ ² . Then in (D[h, T − h], d _∥⋅∥ ) a.s. as n → ∞ we have

$$\left( \hat\rho_{\text{le},\ell}\right)_{t} \longrightarrow (\rho_{\ell})_{t} \qquad\text{and}\qquad (\hat\rho_{\text{ri},\ell})_{t} \longrightarrow (\rho_{\ell})_{t}. $$

Proof

Again we conclude $(1/n){\sum }_{i=1}^{n} \xi _{i}\xi _{i+\ell }\to \mathbb {E}[\xi _{1}\xi _{1+\ell }]$ a.s. as n → ∞. Using N _{n T} → ∞, we find $(1/N_{nt}){\sum }_{i=1}^{N_{nt}} \xi _{i}\xi _{i+\ell }\to \mathbb {E}[\xi _{1}\xi _{1+\ell }]$ a.s. as n → ∞. With a similar argument as in Eq. (23), we find for all 0≤s<t almost surely as n → ∞

$$ \frac{1}{N_{nt}-N_{ns}-(\ell+1)}\sum\limits_{i=N_{ns+2}}^{N_{nt}-\ell} \xi_{i}\xi_{i+\ell}\to\mathbb{E}[\xi_{1}\xi_{1+\ell}]. $$

(25)

Together with the previous Lemma A.4 this implies the almost sure convergence $\hat \rho _{\text {ri},\ell }\to \mathbb {E}[\xi _{1}\xi _{1+\ell }]-\mathbb {E}[\xi _{1}]^{2} = \rho _{\ell }$ for every fixed t and thus for the finite dimensional marginals.

In order to obtain the convergence in (D[h, T − h], d _{∥ ⋅ ∥}), we show a.s. as n → ∞ that

$$ \left( \frac{\mu}{nh}\sum\limits_{i=N_{nt}+2}^{N_{n(t+h)}} \xi_{i}\xi_{i+\ell}\right)_{t} \longrightarrow (\mathbb{E}[\xi_{1}\xi_{1+\ell}])_{t}. $$

(26)

The convergence of the finite dimensional marginals follows from Eq. (25) together with Lemma A.3 and Slutsky’s theorem. We show the uniform convergence (26) even for t ∈ [0, T − h]. It suffices to show almost surely that

$$\begin{array}{@{}rcl@{}} \lim\limits_{n\to\infty} \sup\limits_{t\in[0,T-h]} \frac{\mu}{nh}\sum\limits_{i=N_{nt}+2}^{N_{n(t+h)}} \xi_{i}\xi_{i+\ell} &\le \mathbb{E}[\xi_{1}\xi_{1+\ell}],\\ \lim\limits_{n\to\infty} \inf\limits_{t\in[0,T-h]} \frac{\mu}{nh}\sum\limits_{i=N_{nt}+2}^{N_{n(t+h)}} \xi_{i}\xi_{i+\ell} &\ge \mathbb{E}[\xi_{1}\xi_{1+\ell}]. \end{array} $$

(27)

Again, we make use of a discretization argument as in Messer et al. (2014). We make it explicit here, since the mixing terms ξ _i ξ _{i + ℓ} were not explicitly considered in the latter article. For an ε > 0 with $T/\varepsilon \in \mathbb N$, we decompose the time interval [0, n T] into equidistant sections of length n ε. Using the notation $|\lceil x\rceil |:=\lceil x \rceil +1, x\in \mathbb R,$we bound

$$\begin{array}{@{}rcl@{}} && \sup\limits_{t\in[0,T-h]} \frac{\mu}{nh}\sum\limits_{i=N_{nt}+2}^{N_{n(t+h)}} \xi_{i}\xi_{i+\ell} \\ && \le \max\limits_{j=0,1,\ldots,T/\varepsilon - |\lceil h/\varepsilon \rceil |} \frac{\mu}{nh}\sum\limits_{i=N_{jn\varepsilon}}^{N_{jn\varepsilon+n|\lceil h/\varepsilon\rceil|\varepsilon} } \xi_{i}\xi_{i+\ell}\\ && \le \max\limits_{j=0,1,\ldots,T/\varepsilon - |\lceil h/\varepsilon \rceil |} \frac{\mu}{nh}\sum\limits_{i=N_{jn\varepsilon +nh}}^{N_{jn\varepsilon+n| \lceil h/\varepsilon\rceil | \varepsilon} } \xi_{i}\xi_{i+\ell}\\ &&\qquad + \max\limits_{j=0,1,\ldots,T/\varepsilon - |\lceil h/\varepsilon \rceil |} \frac{\mu}{nh}\sum\limits_{i=N_{jn\varepsilon}}^{N_{jn\varepsilon+nh} } \xi_{i}\xi_{i+\ell}. \end{array} $$

For any δ > 0 we can choose ε > 0 so that maxj=0,…,T/ε−|⌈h/ε⌉|(N _{j n ε + n|⌈h/ε⌉|ε} −N _{j n ε + n h} )/(δ n/μ) → 1 a.s. for n → ∞. Then, for n → ∞, the first summand in the latter display converges to $(\delta /h)\mathbb {E}[\xi _{1}\xi _{1+\ell }]$ a.s. and the second summand to $\mathbb {E}[\xi _{1}\xi _{1+\ell }]$ a.s., since convergence (26) holds for finitely many t. Since δ can be chosen arbitrarily small, we find the first inequality of Eq. (27). The second one follows analogously. Thus, the convergence in Eq. (26) follows. We then exchange the normalization according to Lemma A.3. Omitting ℓ + 1 summands does not change the limit such that the uniform a.s. convergence of $(\hat \rho _{\text {ri},\ell })_{t}$ is shown. Analogously, the uniform a.s. convergence of $(\hat \rho _{\text {le},\ell })_{t}$ is shown. □