Nondifferentiable activity in the brain

Abstract Spike raster plots of numerous neurons show vertical stripes, indicating that neurons exhibit synchronous activity in the brain. We seek to determine whether these coherent dynamics are caused by smooth brainwave activity or by something else. By analyzing biological data, we find that their cross-correlograms exhibit not only slow undulation but also a cusp at the origin, in addition to possible signs of monosynaptic connectivity. Here we show that undulation emerges if neurons are subject to smooth brainwave oscillations while a cusp results from nondifferentiable fluctuations. While modern analysis methods have achieved good connectivity estimation by adapting the models to slow undulation, they still make false inferences due to the cusp. We devise a new analysis method that may solve both problems. We also demonstrate that oscillations and nondifferentiable fluctuations may emerge in simulations of large-scale neural networks.


Fig. S1. Close-up views of cross-correlation histograms obtained with different recording durations.
Cross correlograms of spike trains modulated with (a) smooth oscillations and (b) non-differentiable fluctuations as demonstrated in Figs.2a and 2c.Observation duration  was varied from 1, 10, to 100 hours.Red lines are analytically obtained CCs for individual models, as was obtained in METHODS.

B Autocorrelation histograms obtained with spike time dithering.
To examine whether the central peak of the autocorrelation in Fig. 3b is a non-smooth cusp or it is just a smooth hump, we apply jittering or dithering to a summed spike train so that individual spike times are randomly shifted around the original times (References 2-4), and see how the peak of the autocorrelation histogram collapses according to the dithering.The SDs of the dithering times are 5, 10, and 50 ms.It is observed in Fig. S2 that an apparently non-smooth cusp in the original autocorrelation function collapsed into a smooth hump, even with the short dithering of 5 or 10 ms.This suggests that there is a sharp coherent activity in the brain on the order of a few milliseconds.

C Fitting differentially discontinuous and continuous functions to autocorrelation histograms.
To provide a way to determine non-differentiable activity from discrete time data, which may also be associated with spike sorting errors, we have compared a piecewise differentially discontinuous function with a differentially continuous function for their likelihood or the goodness of fit to the autocorrelation histogram of the summed spike train (with the delta peak that is due to self-spike counts is removed).We have chosen the piecewise differentially discontinuous function as in ShinGLMCC with a coefficient  = 10 8 , while the differentially continuous function as having no break in the prior distribution of the second order derivative.As a result, we have confirmed that the central peak of the autocorrelation histogram may be better represented by a piecewise differentially discontinuous function than by a differentially continuous function.

Fig. S7. Comparison of analysis methods for the posterior region including visual areas. a-c
Connection matrices obtained for biological data using the Classical CC method, the GLMCC method, and the newly developed ShinGLMCC method, respectively.Magenta and cyan squares in each matrix represent excitatory and inhibitory connections, respectively estimated by each analysis method.d Venn diagram representing the relationships between determined connectivities.Sample CCs are displayed along the periphery.

E . Comparison of ShinGLMCC and GLMCC in their goodness of fit to the crosscorrelograms.
To see the difference in the local dynamics of neuron pairs, we have compared ShinGLMCC and GLMCC for their goodness of fit to individual cross-correlograms.Figure S8 shows matrices indicating the neuron pairs for which ShinGLMCC fits the cross-correlogram better (the likelihood is higher) than GLMCC.Here, neurons are sorted as linearly aligned along the Neuropixel.There are significant differences in the cusp cross-correlogram dominance between different brain regions.

F Consistency of identified connections.
We have tested how the excitatory or inhibitory characteristics of identified connections are consistent for individual neurons, by computing the excitatory-inhibitory (E-I) dominance index proposed in Reference (7), defined as  ei = ( e −  i )/( e +  i ) where  e and  i are the numbers of identified excitatory and inhibitory connections projecting from each neuron, respectively.We have shown the distributions of the E-I dominance index in Fig. S9.The Classical CC, GLMCC, and ShinGLMCC gave connections to 241, 189, and 142 neurons out of a total of 242 neurons.The average absolute values of E-I dominance | ei | were 0.78, 0.81, and 0.86, and the fractions of neurons expressing perfect consistency ( ei = 1 or −1) were 0.28, 0.69, and 0.75, respectively, suggesting that ShinGLMCC may have provided a reliable inference.

Fig. S2 .
Fig. S2.Close-up images of autocorrelation histograms of dithered spike times.Autocorrelation histograms were computed for dithered spike trains in which individual spike times were randomly displaced with SDs of 5, 10, and 50 ms.The autocorrelation histograms are plotted for an interval of [-50, 50] ms with a 0.1 ms bin. a Visual cortex, hippocampus, and thalamus (close-up view of Fig. 3b); b Frontal cortex including motor areas; and c Posterior cortex including visual areas.

Fig. S3 .
Fig. S3.Fitting differentially discontinuous and continuous functions to autocorrelation histograms.Piecewise differentially discontinuous functions and differentially continuous functions are fitted to autocorrelation histograms of summed spike trains.The data sets are the same as those in Fig. S2.The likelihood of the differentially discontinuous function is higher than that of the differentially continuous function by 24724 and 28964, and 1917 for a, b, and c, respectively.D Analysis of other data sets recorded in different brain regions.Here we analyze two sets of spike trains in References(5,6) recorded from brain regions that are different from the data analyzed in the main text (the visual cortex, hippocampus, and thalamus).The recorded regions were (A) the frontal cortex including motor areas (Figs.S4 and S5) and (B) the posterior cortex including visual areas (Figs.S6 and S7).

Fig. S4 .
Fig. S4.The population activity of the frontal region including motor areas.a (Left-top) A time histogram for 10 s, with a time bin of 10 ms.(Left-bottom) Spike rasters.(Right-top) A time histogram of a shuffled spike train for 2 s. (Right-bottom) Rasters of shuffled trains.b and c An autocorrelation histogram (1ms bin) and a power spectrum of the summed spike train.

Fig. S5 .
Fig. S5.Comparison of analysis methods for the frontal region including motor areas.a-cConnection matrices obtained for biological data using the Classical CC method, the GLMCC method, and the newly developed ShinGLMCC method, respectively.Magenta and cyan squares in each matrix represent excitatory and inhibitory connections, respectively estimated by each analysis method.d Venn diagram representing the relationships between determined connectivities.Sample CCs are displayed along the periphery.

Fig. S6 .
Fig. S6.The population activity of the posterior region including visual areas.a (Lefttop) A time histogram for 10 s, with a time bin of 10 ms.(Left-bottom) Spike rasters.(Righttop) A time histogram of a shuffled spike train for 2 s. (Right-bottom) Rasters of shuffled spike trains.b and c An autocorrelation histogram (1ms bin) and a power spectrum of the summed spike train.

Fig, S8 .
Fig, S8.Matrices indicating the neuron pairs for which ShinGLMCC better fits the cross-correlograms.a Visual cortex, hippocampus, and thalamus (close-up view of Fig. 3b); b Frontal cortex including motor areas; and c Posterior cortex including visual areas.