Existence of anticorrelations for local field potentials recorded from mice reared in standard condition and environmental enrichment

In the present paper, we analyze local field potentials (LFPs) recorded from the secondary motor cortex (M2) and primary visual cortex (V1) of freely moving mice reared in environmental enrichment (EE) and standard condition (SC). We focus on the scaling properties of the signals by using an integrated approach combining three different techniques: the Higuchi method, detrended fluctuation analysis, and power spectrum. Each technique provides direct or indirect estimations of the Hurst exponent $H$ and this prevents spurious identification of scaling properties in time-series analysis. It is well known that the power spectrum of an LFP signal scales as $1/f^{\beta }$ with $\beta >0$. Our results indicate the existence of a particular power spectrum scaling law $1/f^{\beta }$ with $\beta <0$ for low frequencies ($f<4$ Hz) for both SC and EE rearing conditions. This type of scaling behavior is associated to the presence of anticorrelation in the corresponding LFP signals. Moreover, since EE is an experimental protocol based on the enhancement of sensorimotor stimulation, we study the possible effects of EE on the scaling properties of secondary motor cortex (M2) and primary visual cortex (V1). Notably, the difference between Hurst's exponents in EE and SC for individual cortical regions (M2) and (V1) is not statistically significant. On the other hand, using the detrended cross-correlation coefficient, we find that EE significantly reduces the functional coupling between secondary motor cortex (M2) and visual cortex (V1).

Figure 1

Integrated approach applied to signals generated using the Fourier filtering method, with given Hurst's exponents. In the top-left panel $\langle L\left(k\right)\rangle$ against $k$ is plotted in a double-logarithm scale. In the top-right panel $\ln \left(F\right(n\left)\right)$, obtained through the DFA, is plotted against $\ln \left(\mathrm{n}\right)$ in a double-logarithm scale. The corresponding DFA exponent $\alpha$ is in keeping with Hurst's exponent estimated using the fractal dimension,  i.e., $\alpha =0.69$ and $\alpha =0.31$. In the bottom panels the corresponding power spectra in a double logarithm scale are shown. For the case $H=0.7$ and $H=0.3$ the linear fitting gives $\beta =0.41$ (bottom left) and $\beta =-0.39$ (bottom right), respectively.

Figure 2

DFA applied to envelope $a\left(t\right)$ [see Eq. (28)] of the artificial FGn signals, generated using the Fourier filtering method with given Hurst's exponents: specifically $H=0.3$ ($H=0.7$) for the top (bottom) curve. The fluctuation, obtained through the DFA, against $n$ in a double-logarithm scale is shown. The corresponding DFA exponents $\alpha$ are in disagreement with Hurst's exponents of the whole signals,  i.e., $\alpha =0.51$ ($\alpha =0.56$) for the top (bottom) curve.

Figure 3

(Left) LFP signal from visual cortex (V1) of an animal in SC. (Right) In the double-logarithm diagram the spectrum of the LFP (left figure) has typical features. The plot qualitatively shows an increasing $\beta <0$ and decreasing $\beta >0$ behavior of the spectrum depending on frequency intervals (the frequency $f_k$ is expressed in $\mathrm{Hz}$). Specifically, two frequency intervals where the spectrum can be well approximated by a linear fit can be identified. The intervals are identified by the shaded areas and represent two well-defined physiological frequency bands, the delta band $f<4$ Hz and beta band 13–30 Hz.

Figure 4

Integrated approach applied to a LFP signal from visual cortex (V1) of an animal in SC (see Fig. 3). In the top panel is shown the length of the fractal curve, calculated through the Higuchi method, against $k$ in a double-logarithm scale. The fractal dimension $D$ turns out to be $D\pm \Delta D=1.77\pm 0.01$. In the middle panel the results obtained through DFA are shown. The corresponding value of the DFA exponent is $\alpha \pm \Delta \alpha =0.22\pm 0.01$ and is consistent with the value of Hurst's exponent calculated through the fractal dimension $H_D=2-D=0.23\pm 0.01$. In the bottom panel, the power spectrum $S\left(f_k\right)$ of the signal against $f_k$ is shown in a double-logarithm scale (the frequency $f_k$ is expressed in $\mathrm{Hz}$). The corresponding value of $\beta$ is $\beta \pm \Delta \beta =-0.7\pm 0.1$ and the corresponding value of Hurst's exponent is $H_{\beta }=\frac{\beta +1}{2}=0.15\pm 0.05$.

Figure 5

Determination of the scaling intervals and scaling exponents of fractal dimension, DFA and power spectrum techniques applied to a LFP signal from visual cortex (V1) of an animal reared in SC (see Fig. 4). In the top panel are shown the local slopes $\theta {\left(k\right)}_{\mathrm{fd}}$ of the length of the fractal curve $\langle L\left(k\right)\rangle$. The scaling interval falls in the region $I_{\ln \left(k\right)}=[1.7,3.7]$, where the mean local slope is equal to ${\theta }_{\mathrm{fd}}\pm \Delta {\theta }_{\mathrm{fd}}=-1.77\pm 0.01$. In the middle panel the results obtained for the DFA are shown. The corresponding scaling region is determined by the interval $I_{\ln \left(n\right)}=[3.3,4.3]$ in which the mean local slope assumes the value ${\theta }_{\mathrm{dfa}}\pm \Delta {\theta }_{\mathrm{dfa}}=0.22\pm 0.01$. In the bottom panel, the scaling region is $I_{\ln \left(f_k\right)}=[-0.3,0.65]$ and is characterized by values of the mean local slope ${\theta }_{\mathrm{ps}}\pm \Delta {\theta }_{\mathrm{ps}}=0.7\pm 0.1$ (the frequency $f_k$ is expressed in $\mathrm{Hz}$).

Figure 6

Integrated approach applied to LFP signal from secondary motor cortex (M2) of a mouse reared in EE. In the top panel is shown the length of the fractal curve, calculated through the Higuchi method, against $k$ in a double-logarithm scale. The local slopes give the fractal dimension $D\pm \Delta D=1.83\pm 0.01$. In the middle panel the results obtained through DFA are shown. The value of the DFA exponent is $\alpha \pm \Delta \alpha =0.17\pm 0.01$ and is consistent with the value of Hurst's exponent calculated through the fractal dimension $H_D=2-D=0.17\pm 0.01$. In the bottom panel, the power spectrum $S\left(f_k\right)$ of the signal against $f_k$ is shown in a double-logarithm scale (the frequency $f_k$ is expressed in $\mathrm{Hz}$). The corresponding value of $\beta$ is $\beta \pm \Delta \beta =-0.3\pm 0.1$ and the corresponding value of Hurst's exponent is $H_{\beta }=\frac{\beta +1}{2}=0.35\pm 0.05$.

Figure 7

Determination of the scaling intervals and scaling exponents of fractal dimension, DFA, and power spectrum techniques applied to a LFP signal from secondary motor cortex (M2) of a mouse reared in EE (see Fig. 6). In the top panel are shown the local slopes $\theta {\left(k\right)}_{\mathrm{fd}}$ of the length of the fractal curve $\langle L\left(k\right)\rangle$. The scaling region is located in the interval $I_{\ln \left(k\right)}=[2.0,4.2]$ where the mean local slope assumes a value ${\theta }_{\mathrm{fd}}\pm \Delta {\theta }_{\mathrm{fd}}=-1.83\pm 0.01$. In the middle panel the results obtained for the DFA are shown. The corresponding scaling interval falls within the region $I_{\ln \left(n\right)}=[3.4,4.5]$ determined by value of the mean local slope equal to ${\theta }_{\mathrm{dfa}}\pm \Delta {\theta }_{\mathrm{dfa}}=0.17\pm 0.01$. In the bottom panel the scaling region for the power spectrum $S\left(f_k\right)$ of the signal is determined. The scaling interval turns out to be $I_{\ln \left(f_k\right)}=[-1,0]$ and is characterized a mean local slope ${\theta }_{\mathrm{ps}}\pm \Delta {\theta }_{\mathrm{ps}}=0.3\pm 0.1$ (the frequency $f_k$ is expressed in $\mathrm{Hz}$).

Figure 8

Integrated approach applied to a surrogate LFP signal from secondary motor cortex (M2) of an animal reared in EE. In the top panel the length of the fractal curve, calculated through the Higuchi method, against $k$ in a double-logarithm scale, is shown. The fractal dimension $D$ turns out to be $D=1.5$. In the middle panel the results obtained through DFA are shown. In the bottom panel it is shown the power spectrum $S\left(f_k\right)$ of the signal against $f_k$ in double-logarithm scale (the frequency $f_k$ is expressed in Hz).

Figure 9

Plots of the DCCA coefficients of representative SC and EE animals. In both cases ${\rho }_{\mathrm{DCCA}}$ assumes nearly constant values for $n>3000$. The plots of the mean ${\rho }_{\mathrm{DCCA}}$'s computed from the surrogate data of the LFP signals are also shown. For each mean ${\rho }_{\mathrm{DCCA}}$ the corresponding standard errors (shadowed area) are displayed. As expected, the surrogate DCCA coefficients oscillate near zero for large-enought $n$ and are much smaller than the asymptotic value for the corresponding LFP signals.