Characteristics of temporal fluctuations in the hyperpolarized state of the cortical slow oscillation

We present evidence for the hypothesis that transitions between the low- and high-firing states of the cortical slow oscillation correspond to neuronal phase transitions. By analyzing intracellular recordings of the membrane potential during the cortical slow oscillation in rats, we quantify the temporal fluctuations in power and the frequency centroid of the power spectrum in the period of time before “down” to “up” transitions. By taking appropriate averages over such events, we present these statistics as a function of time before transition. The results demonstrate an increase in fluctuation power and time scale broadly consistent with the slowing of systems close to phase transitions. The analysis is complicated and limited by the difficulty in identifying when transitions begin, and removing dc trends in membrane potential.

Figure 1

(Color online) A cortical pyramidal neuron recorded intracellularly during the experiments from which the sample is taken. The basal dendrites (double arrowhead) arising from the pyramidal soma $\left(S\right)$ are studded with dendritic spines, as are the oblique branches (enlarged inset) arising from the apical dendrite (arrowhead). The membrane potential (“Vm” in top inset) responds nonlinearly to negative current pulses (“Im”) injected into the soma and fires a single action potential to a just-threshold positive current pulse.

Figure 2

(a) The membrane potential against time. The points of transition as identified by the threshold are marked with a “$+$” symbol. (b) A close-up view of two of the down-to-up transitions. (c) The second derivative of the membrane potential against time; the first-derivative being evaluated with a center-difference method based on 50 data points (or $0.005\mathrm{s}$) forward and backward of the point of interest; the second-derivative using 100 data points $\left(0.01\mathrm{s}\right)$ forward and backward. The extremely high values of the second derivative are due to action potentials, as can be seen from comparison with part (a). (d) A close-up view of the second derivative, for the same section of data as part (b). The threshold of $12500\mathrm{mV}∕{\mathrm{s}}^2$ marked with a dashed line. The points of transition are indicated with the vertical bars. The second derivative must remain below the threshold for a period of more than $0.2\mathrm{s}$ for the upward crossing to be accepted as a transition.

Figure 3

(Color online) (a) An example sequence of membrane potential against time. For clarity, only $15\mathrm{s}$ out of $90\mathrm{s}$ has been shown. The points that have been identified as transitions are marked by crosses. A close-up view of one of the transitions is shown. (b) The mean membrane potential during the $0.25\mathrm{s}$ period before a transition, for the neuron shown in sequence (a). The average has been taken over all the identified transition periods. The $x$ axis indicates time before transition—i.e., $t=0\mathrm{s}$ denotes the transition point and $t=-0.25\mathrm{s}$ denotes $0.25\mathrm{s}$ before the transition; the minus sign is introduced to enable the graph to be read left to right. The gray lines indicate one standard deviation either side of the mean. (c) The membrane potential minus the mean membrane potential, for the $0.25\mathrm{s}$ periods before some of the transitions. (For clarity, not all the series are shown.) Note how there is a wider spread in the sequences at the transition point $(t=0)$ compared to $0.1\mathrm{s}$ previously $(t=-0.1\mathrm{s})$. The top arrow signifies the length of the sliding window that has been used to analyze the frequency data; the bottom arrow shows the time before the transition.

Figure 4

(Color online) (a) A plot of the power spectrum for one neuron. The five lines denote the mean power over $0.050\mathrm{s}$ segments that are centred at $0.025\mathrm{s}$ (black), $0.050\mathrm{s}$ (blue), $0.075\mathrm{s}$ (green), $0.100\mathrm{s}$ (red), and $0.125\mathrm{s}$ (yellow) before the transition. The mean is taken over all the identified transitions in the sequence for this neuron. The dashed lines either side of the $0.025\mathrm{s}$ line indicate the standard uncertainty in this result. (The standard uncertainties for the other lines are similar in size.) Also shown by the heavy dashed line is a curve of the form $S\left(f\right)=B+A∕f^2$, where $S$ is power, $f$ is frequency, and $A$ and $B$ are constants fitted by eye. (b) The power spectrum minus $S\left(f\right)$, for the neuron of part (a). Here we see the increase in power at low frequencies as the transition is approached.

Figure 5

(a) The mean membrane potential in the $0.25\mathrm{s}$ period before a down to up transition, averaged over all transitions used in this study, i.e., ${\sum }_{k=1}^M{P}^k{\overline{y}}^k\left(j\right)∕{\sum }_{k=1}^M{P}^k$. Note that it rises on the approach to transition $(t=0)$. Negative times denote the time before transition. (b) The mean fluctuation power as a function of time before transition. Note that only frequencies below $500\mathrm{Hz}$ have been used in this calculation. Averages have been taken over all sequences and transitions. The dashed lines indicate the standard uncertainty in the mean. (c) The centroid frequency of the power spectrum, averaged over all sequences used in this study, as a function of time before transition. The dashed lines indicate the standard uncertainty in the mean.