# Symmetry making and symmetry breaking in cortex
A collective portrait of ensemble excitation and inhibition

Nima Dehghani<EMAIL_ADDRESS>McGovern Institute for Brain Research,
MIT, Cambridge, MA  Department of Physics, MIT, Cambridge, MA

###### Abstract

Creating a quantitative theory for the cortex poses several challenges and
raises numerous questions. For instance, what are the significant scales of
the system? Are they micro, meso or macroscopic? What are the relevant
interactions? Are they pairwise, higher order or mean-field? And what are the
control parameters? Are they noisy, dissipative or emergent?

To tackle these issues, we suggest using an approach similar to the one that
has transformed our understanding of the state of matter. This includes
identifying invariances in the ensemble dynamics of various neuron functional
classes, searching for order parameters that connect important degrees of
freedom and distinguish macroscopic system states, and identifying broken
symmetries in the order parameter space to comprehend the emerging laws when
many neurons interact and coordinate their activation.

By utilizing multielectrode and multiscale neural recordings, we measure the
scale-invariant balance between excitatory and inhibitory neurons. We also
investigate a set of parameters that can assist us in differentiating between
various functional system states (such as the wake/sleep cycle) and
pinpointing broken symmetries that serve different information processing and
memory functions. Furthermore, we identify broken symmetries that result in
pathological states like seizures.

## Introduction

Over the past two decades, advances in microelectronic fabrication techniques
have made it possible to record the electrical activity of large populations
of neurons both _in vitro_ and _in vivo_. With the increasing size of these
populations, researchers have shifted their focus from studying the responses
of individual neurons to stimuli (such as in the seminal works of [1, 2]) to
examining population dynamics and correlations between neurons [3, 4, 5],
pairwise and/or higher-order correlations [6, 7, 8, 9, 10]. Despite the
general agreement that information is represented and processed through
correlated activity patterns in neuron populations, descriptive statistical
models fail to provide significant theoretical insights into the collective
behavior of these populations. As a result, attempts to find principles, while
acknowledging high variability in spiking patterns and spontaneous activity as
minimum theoretical constraints, adopt a top-down compartmentalization
approach [11]. This approach follows Marr’s tri-level hypothesis [12, 13] and
separates the problem into a computational (behavioral) level, an algorithmic
level, and a biological implementation level. Alternatively, some suggest a
quantitative computational framework that starts from the tri-level approach
and adds further requirements such as hierarchical representation and learning
[14]. However, these approaches have their limitations. Their assumption of
separation of scales is at odds with the nature of complex systems [15, 16].
To describe hierarchical representation that matches the nature of the cortex
as an adaptive physical system, a successful theory must account for the
interaction between sub-assemblies while disregarding insignificant details in
transferring dynamics between observation scales [16, 17].

Given the challenges in modeling the activity of large neural populations, it
is crucial to search for signatures that can guide us in formulating an
effective theory that describes the macroscopic characteristics of the system,
despite the high degrees of freedom at the microscopic scale of individual
neurons. As the number of measured neuronal activity increases linearly, the
number of interactions among neurons grows exponentially, making it difficult
to obtain a representative sample of the full distribution of activity
patterns through correlation measures or dimensionality reduction of
population dynamics. This is where physics can help if we look for collections
of patterns and symmetries to find order parameters and broken symmetries that
point to underlying laws [18, 19]. Instead of solely focusing on cognition or
animal behavior as metrics for describing the system, we should identify
invariances in ensemble dynamics, search for order parameters that link
important degrees of freedom that differentiate macroscopic states of the
system, and identify broken symmetries in the order parameter space to
understand the laws that emerge when many neurons interact and orchestrate
their activation.

Recently, we have shown invariance of E:I balance across scales [20]. However,
in searching for the right order parameters, we have also found that a maximum
entropy model of weak pairwise interactions (Ising model) provides a good fit
for inhibitory neurons but fails to describe the spiking patterns of
excitatory neurons [21]. The importance of diversity in computing node classes
becomes evident when E:I balance breakdown leads to seizure, a distinctive
pathological state where information processing capacity is near zero while
macroscopic neuronal activity shows high levels of synchronicity [20].
Additionally, thermodynamic-based measures of population activity (e.g. heat
capacity) fail to differentiate macroscopic states of the system (i.e. awake,
light sleep, deep sleep) [21]. Here, we extend these studies to further
investigate multiscale properties, invariance of E:I balance, search for order
parameters that differentiate states of the system, and examine the link
between microscopic and macroscopic features of pathological symmetry breaking
during seizure.

Figure 1:  Data. _Top._ The implant location (left) and multielectrode array
(right). The 10x10 array covers an area with the span of 4mm x 4mm. _Bottom._
Sample recording from the multielectrode array.

## results

Understanding cortical neural activity presents the challenge of dealing with
the multiple spatial and temporal scales inherent in neural dynamics. This
multiscale neural activity exhibits behaviors that resemble state transitions
in other physical systems with dynamics that transcend many different length
scales [22]. The renormalization group provides a powerful framework for
studying symmetries and changes in a physical system as viewed at different
scales [23, 24]. In the context of phase transitions, renormalization group
theory is based on the idea that only slow modes are responsible for the phase
transition, and phase transitions are closely tied to long-range behavior [25,
26]. The defining property of criticality, according to renormalization group
theory, is that the characteristic length scale of the structure of the
physical system, also known as the correlation length, becomes infinite [27].
Similarly, large ensembles of neural events provide data for describing long-
range coupling in the temporal domain, comparable to spatial coupling in phase
transitions. Information in the system is conveyed by the temporal order of
neural packets and different collective states emerge as the functional
network changes. Thus, our study relies on multiscale dynamical features of
ensemble E:I balance across different states of the wake/sleep cycle to better
understand collective neural activity (see Fig.1 and Fig.2).

###  .1 Multiscale Signatures of Balance Invariance

####  .1.1 Excitation-Inhibition Balance Across Temporal Scales

Recently, we have shown that across all different states of the wake-sleep
cycle, excitatory and inhibitory ensembles are well balanced, and co-fluctuate
with slight instantaneous deviations from perfect balance, mostly in slow-wave
sleep. Remarkably, these correlated fluctuations are seen for many different
temporal scales. The similarity of these computational features with a network
model of self-generated balanced states suggests that such balanced activity
is essentially generated by recurrent activity in the local network and is not
due to external inputs [20]. To further examine invariance, we tested balance
across multiple temporal scales ranging from 1 millisecond to 10 seconds (see
Fig.2 inset). When the excitatory and inhibitory ensemble activities are
normalized, they appear as scaled mirror images of each other. This invariance
was observed in both awake and different sleep states (i.e. slow-wave sleep,
light sleep, and rapid-eye movement sleep). As an example, a multi-step zoom
across scales (Fig.2) shows that when the firing of a given functional
category is normalized to the total related ensemble firing power, Excitation
and Inhibition mirror each other’s fluctuations en-large. It is important to
note that with any current technology for ensemble recordings, we are
dramatically subsampling the possible space of neurons and can only extract
estimates of balance rather than its exact magnitude. The true scaling of
balance could only be calculated if activity of all excitatory and inhibitory
neurons were measured, a challenge not met with current technology but perhaps
achievable in the far future.

Figure 2:  Multiscale representation of balanced Excitation and Inhibition
_Ensemble spiking Excitation-Inhibition balance._ Sample ensemble spiking with
different levels of coarse-graining are shown in the four color panels (Note,
these panels are example subsets from the rastergram and LFP shown in Fig.1).
Preservation of Excitation-Inhibition balance across scales shows mirrored
activity for different lengths of times (scales). The negative sign is only
used conventionally to represent the opposing nature of Inhibition (red)
versus Excitation (blue). In each panel, the ensemble firing of a given cell
category is normalized by the total firing power in the same category during
the shown epoch. Each panel has 656 bins and the duration of each bin is shown
in the middle panel. For example, the top right panel represents scale 30 and
a duration of 1.0938 hr, showing 656 bin’s duration of 6003(sec). The mean
(green) and standard deviations (magenta) of each epoch are shown with dashed
horizontal lines, respectively. Vertical bars show the time window from which
another coarser scale 656-bin epoch was chosen. The middle panel shows the
even logarithmic spacing of scales, which was chosen for computational
efficiency.

####  .1.2 Scale-Invariance in Correlated Fluctuations of Excitation and
Inhibition in the Cerebral Cortex

A common pattern we observe is that the ensemble E-I distribution’s standard
deviation (and variance) shrinks around the fixed mean as we go from finer to
coarser scales (see Fig.2 panels and Fig.3A). If we look at the variance
around the mean (i.e., coefficient of variation) for the entire recordings, we
observe a symmetric decrease of coefficient of variation across scales
(Table1). This symmetry is lost when we randomize and disrupt the temporal
relation between excitatory and inhibitory ensembles (Fig.3B,C). This
symmetric change of excitatory-inhibitory fluctuations at different scales is
linked to a unidirectional symmetry of excitatory-inhibitory ensemble pairs
that align with the ideal line of balance (the diagonal in Fig.3A represents
the ideal line of balance). The dispersion from the diagonal line of balance
shows a balanced instantaneous dominance of excitatory or inhibitory systems.
This constitutes as an evidence for scale-invariance in terms of correlated
fluctuations of excitation and inhibition in the cerebral cortex. Across
sleep-wake cycle, a clear symmetry is observed across multiple scales. The
Mean absolute value deviation (MAD),
$\frac{1}{n}\sum_{i=1}^{n}|x_{i}-\overline{X}|$, (a statistical measure of
dispersion) for different states for awake, slow-wave sleep (SWS), Light sleep
(LSL) and Rapid-eye movement (REM) had similar structure and statistical
dispersion (see Table1), We measured the dispersion, which is the distance of
each point from the diagonal line of equal excitatory and inhibitory activity,
in a scatter plot of the ratio of excitatory and inhibitory activity for each
pair of neurons. The dispersion indicates how much one functional system
(Excitation or Inhibition) prevails over the other. We found that the
dispersion decreased across all states (awake, light sleep and slow-wave
sleep) as we increased the timescale of our analysis, meaning that the
excitatory and inhibitory activity became more balanced. However, the shape of
the scatter plot remained similar even at very high timescales, meaning that
the excitatory and inhibitory activity kept its features at coarser temporal
scales.

Table 1:  Indices of dispersion | MAD111Mean absolute value deviation, $\frac{1}{n}\sum_{i=1}^{n}|x_{i}-\overline{X}|$ | Skewness222$\gamma=\operatorname{E}\Big{[}\big{(}\tfrac{X-\mu}{\sigma}\big{)}^{\\!3}\,\Big{]}$ |  $S_{CV}$333$CV={\sigma}/{\mu}$444$S_{CV}$:Slope of multiscale E:I Coefficient of Variation
---|---|---|---
AWAKE | $0.0023\pm 0.0030$ | $-0.013\pm 0.046$ |
REM | $0.0028\pm 0.0032$ | $0.052\pm 0.015$ | $|S_{CV}-1|<\epsilon$
LS | $0.0021\pm 0.0024\ $ | $-0.010\pm 0.015$ | $\epsilon<10^{-2}$
SWS | $0.0027\pm 0.0034$ | $-0.012\pm 0.087$ |

In both slow-wave sleep and wakefulness, the two systems have a symmetric
distribution of magnitude and frequency of the ensemble fraction of excitation
and inhibition. The cross-subject average skewness (which is a measure of how
asymmetric a distribution is) of excitatory/inhibitory dominance, for
different states across the wake-sleep cycle were close to zero (indicating
perfect symmetry). For details, see Table.1. This symmetry indicates that the
balance is maintained across the scales. As expected, as we move to the coarse
temporal scales, the distribution narrows while still keeping the E:I balance.
The balance of excitatory and inhibitory neurons seems to be a key feature of
cortical computation, but there is a wide parameter space where input can be
computationally processed while preserving balance [28]. In fact, it has been
shown that spike timing is finely tuned by dynamic interactions between
excitatory and inhibitory conductances [29, 30]. Some simulations have even
suggested that the spatial (proximal vs distal) distribution of inhibitory
synaptic activity on pyramidal cells keeps the system more under the control
of inhibition but in an excitable mode waiting for the next burst of
excitation from the thalamus [31].

####  .1.3 Disrupting Temporal and Spatial Structure of Ensemble Spike
Patterns through Randomization

We examined how randomizing the spike patterns of the neurons affected the
multiscale balance of excitatory and inhibitory activity. We used two
different randomizations: one that permuted the inter-spike intervals in the
ensemble series, and one that circularly shifted the inhibitory and excitatory
ensembles (for details, see the methods section). Both randomizations broke
the alignment of excitatory-inhibitory pairs along the diagonal line, which
indicated the balance between the two systems. Instead, randomizations
transformed the data to a circular cloud, which indicated noise. This
resultant “sphering” is similar to a “whitening” transformation because it
turned the input vector into a white noise vector, with no remaining
correlation or structure, as shown in Fig.3A2,A3. Note that the linear
whitening transforms (such as Cholesky, PCA or ZCA) assume that the input
vector has a multivariate normal distribution and use a linear transformation
to make the covariance matrix an identity matrix. In contrast, nonlinear
whitening transforms (implemented through neural networks, kernel whitening or
ICA) can handle non-Gaussian distributions and use a nonlinear transformation
to remove higher-order dependencies.

Each of these randomizations were designed to achieve a specific decorrelation
objective. In “random permutation”, we first calculated the inter-spike
interval (ISI) for the pooled ensemble series of each (excitatory or
inhibitory) functional category, and then randomly shuffled the ISI for that
category. We then reconstructed a new temporal order of ensemble spikes by
cumulatively summing the shuffled ISI. This process preserves the number and
distribution of spikes, but changes their arrangement within the series. In
“circular shift of spike ensemble”, we first calculated the inter-spike
interval (ISI) of each unit’s spike series, and then shifted the spikes of
each unit by a random value between 1 and the maximum ISI of that unit (Note,
here by unit we are referring to a single neuron). The randomized units were
then aggregated to create the randomized ensemble series of each (excitatory
or inhibitory) functional category. This process preserves the internal spike
timing of each unit, but disrupts the timing between units in that category.
The first method, more or less, acts like a linear whitening since it only
disrupts the temporal order of aggregate spikes with respect to each other. In
contrast, the later method, acts closer to a nonlinear whitening since it
decorrelates the ensemble through disrupting inter-unit timing of spikes, thus
removing higher-order correlations. Regardless, both methods of randomization
disrupt the temporal or spatial structure of the ensemble spike patterns of
the neurons.

Figure 3:  Randomization of ensemble activity. _A: Data._ Scatterplot of
normalized ensemble excitation and inhibition. Blue, red, green and black
represent 4 sample scales. Paired E-I scatter as elipsoids with their major
axis of variation along the diagonal (perfect symmetry). _B:Random permutation
of ISI in the ensemble series._ _C:Fixed-ISI circular shift of spikes._
Ensemble Excitation and Inhibition show correlated and balanced structure that
is disrupted by sphering randomization. For details of the randomization
techniques, see methods.

####  .1.4 Limitations of Current Experimental Techniques

It is worth mentioning that the multielectrode array samples spiking activity
from an area about the size of a cortical column using a 10x10 electrode grid.
Although the density of neurons in a cortical column varies across species
[32, 33], the Utah array and other current experimental techniques massively
subsample from the neurons in a cortical column, estimated to range between
10,000 to 20,000 per column [34, 35]. As the ability to record electrical
activity of neurons scales up, we will have the opportunity to search for
translational symmetry of excitatory-inhibitory balance across many
neighboring columns. Given the underlying functional anisotropy at different
spatial scales, such as our prior observation of stereotypical directionality
of local field potential traveling waves at the scale of a cortical column
[36] and in theoretical models of propagating waves across much larger patches
of cortex [37], the existence or lack thereof of ensemble excitatory-
inhibitory translational symmetry will guide us in developing a formalism that
could link micro and macroscales.

###  .2 Universality of Balance

####  .2.1 Collapse Curve Analysis of Ensemble Excitation-Inhibition
Fluctuations

We then investigated multiscale properties of fluctuations of cortical
ensemble excitation-inihibition through rescaling. We constructed a collapse
curve analysis to reveal the self-similar or self-affine properties of these
fluctuations across different time scales and different states of wake-sleep
cycle. A collapse curve can be obtained by rescaling the axes of different
curves of a physical quantity measured at different values of a control
parameter, such that they collapse onto a single curve. This indicates that
the system exhibits some kind of universality or self-similarity. We applied
this method to the observed fluctuations of ensemble activity, represented by
$f_{obs}=Ens_{E}-Ens_{I}$, where $Ens_{E}$ and $Ens_{I}$ are the ensemble
activities of excitatory and inhibitory neurons, respectively. After sorting
$f_{obs}$ in ascending order, we add up all the values from the smallest to
the largest, resulting in a symmetric parabolic curve that reflects the
overall balance of ensemble excitation and inhibition (Fig.4). To test if the
system has self-similar or self-affine properties, we rescale the curve for
each time scale so that it fits between 0 and 1 (x-axis normalization) along
with normalizing the y-axis by the product of the number of bins. This would
allow a reliable comparison of timeseries of different lengths (for example a
longer awake versus a shorter REM recording). We then systematically identify
non-collapsible regions for various comparisons:i) data against random series,
ii) comparison of different scales, and iii) comparison of different states
across various scales. As shown in Fig.4A, surrogates obtained from various
randomization methods behave similar to each other and do not collapse on the
data curve. This observation further signifies that the observed balance is
not merely a meaningless statistical attribute of the recorded timeseries but
rather it is related to a collective pattern of ensemble spiking. Comparing
different scales, Fig.4B, shows that at coarsest scale, surrogate collapse
curve approaches the data collapse curve. Therefore, we can deduce that at the
very long timescales, while balanced excitation and inhibition is still a
property of the system, the packet contents may no longer hold any meaningful
structure from an information processing point of view. Extending this
analysis to comparison of different states across the wake-sleep cycle, shows
that (more or less) different states collapse onto the same regime Fig.4C,
further emphasizing the presence of an invariant balance across scales.

On a corollary note, we wish to emphasize that scaling ansatz can help to
reveal the universal features of a system and to derive scaling relations
among different observable. When different curves of a physical quantity
measured at different values of a control parameter can be collapsed onto a
single curve by rescaling the axes, this behavior is a reflection of scaling
behavior of the system near a critical point. Therefore the scaling collapse
that we examined here threads with other attributes of collective behavior,
namely neural avalanches. Neural avalanches are spatiotemporal patterns of
neuronal activity that occur spontaneously in cortical networks and exhibit
scale-free behavior, meaning that they lack a characteristic size or duration.
Neural avalanches are often considered as a signature of criticality in the
brain, a state where the system is poised at the edge of a phase transition
between order and disorder. Criticality is thought to confer optimal
computational properties to the brain, such as high sensitivity, dynamic
range, and information processing capacity. However, there is an ongoing
debate about whether neural avalanches are truly indicative of criticality or
are simply a consequence of other factors, such as network topology, synaptic
plasticity, or external inputs. Through rigorous statistical measures, we have
revealed that neural avalanches are not well described by self-organized
criticality (SOC) [38], which is a mechanism by which systems can
spontaneously tune themselves to criticality without external tuning
parameters. We argued that SOC models fail to account for the role of
inhibition in shaping neural avalanches and for the diversity of
spatiotemporal patterns observed in cortical networks. In addition, we have
also shown that maximum entropy models, which are statistical models that
capture the pairwise correlations between neurons while maximizing their
entropy, are not good at capturing the intricate dynamics of excitation and
inhibition [21]. Perhaps together, the scaling collapse that we report here
along with the differential behavior of Excitatory and Inhibitory populations
that we have reported in the past, reveal a different type of criticality in
cortical networks. This means that the system exhibits scale-invariant
behavior over a wide range of parameters, rather than at a single critical
point. This could be due to the presence of multiple competing interactions or
feedback mechanisms in cortical networks, which could create multiple phases
or coexisting phases near criticality.

Figure 4:  Fluctuation collapse. _A. Fine-scale collapse curve versus random
surrogates._ The inset shows the histogram distribution of fluctuation of
ensemble activity at the finest scale. _B. Comparison of collapse curves
across different scales._ Note that at coarsest scale, surrogate collapse
curve approaches the data collapse curve. Different random surrogates collapse
onto the same curve but distinctively are different from data collapse curve.
_C. Comparison of collapse curves for different states (SWS: slow-wave sleep,
REM: Rapid Eye Movement, and Awake) across various scales._ Dashed circles are
visual guides to exemplify that more or less collapse curves of different
states have similar behavior in a given scale.

####  .2.2 Partition Curve Analysis of Neuronal Firing Renewal

A Poisson process is a simple way to describe neuronal firing as a continuous-
time Markov process with independent and identically distributed holding times
where the inter-arrival times are exponentially distributed [39, 40]. However,
this model cannot accurately describe realistic interspike interval
distributions because it does not account for neuronal refractoriness, which
can lead to periods of time during which a neuron is unable to fire another
spike [41, 42, 43]. This can result in gaps in the ensemble spike timeseries,
making it difficult to accurately model the behavior of the ensemble.
Refractoriness, particularly during in vivo high conductance state, plays a
significant role in the variability the observed neuronal responses [44]. To
account for refractoriness, and the resultant neural response variability in
the stochastic description of spike trains, filtering or more sophisticated
models such as renewal or Hawkes processes can be used [45]. Each of these
methods have certain drawbacks; for example, filtering can remove short-term
fluctuations but also introduces bias. Renewal theory extends the Poisson
process to include arbitrary holding times and is an idealized model for
random events, with the assumption that the times between events are
independent and identically distributed. Renewal processes keep a memory of
the last event but not of any earlier events and can be used to model ensemble
spiking patterns by taking into account the time since the last event.
However, renewal processes assume that inter-arrival times are independent and
identically distributed, which may not hold for nonstationary data that often
exhibits trends or other patterns that violate this assumption [46, 45, 47].
Spike trains of typical neurons in vivo show irregular behavior, with large
variability in interspike intervals and spike counts [48]. Whether this
irregularity is due to thermal noise, microscopic chaos, or an intricate
neural code is still an open question [49, 50]. Apart from intrinsic noise
sources at the level of an individual neuron, there are also extrinsic noise
sources due to signal transmission and network effects (extrinsic noise).
Variability can be quantified by the coefficient of variation value of
interval distributions [51, 52]. In neurons with strong adaptation, interspike
intervals are not independent rendering the renewal theory as a weak model. A
time-dependent generalization of renewal theory can be used to account for
non-stationary real-world stimuli with a strong temporal component. Hawkes
processes, which are self-exciting and can model refractoriness, can be used
for both stationary and non-stationary data, but there are restrictions on
their use for non-stationary data. If the Hawkes process intensity function
varies with time, the model is no longer stationary [53]. To examine the
temporal variation of the probability of an event occurring at a given time we
need new methods.

We present a method to analyze the distribution of ensemble activity in a
given functional category using a partition curve. The partition curve
represents a size-based distribution that indicates the percentage of ensemble
activity of a certain size or in a certain size interval. The shape of the
partition curve can provide information about the equivalence or unequivalence
of the distribution of ensemble activity. This method, in conjunction with
Hawkes theory, could potentially be used to analyze ensemble spiking patterns
in neurons. Analysis of the distribution of ensemble activity using a
partition curve can help us to gain a deeper understanding of the relationship
between ensemble spiking patterns and renewal processes. Figure 5 shows the
probability–probability plot comparing the distribution of a variable against
a hypothetical uniform distribution of that variable. A partition curve is a
graphical representation of the distribution of ensemble spike firing rate
that plots the cumulative percentage of the variable against the cumulative
percentage of the population. It represents a size-based distribution that
indicates the percentage of ensemble activity of a certain size or in a
certain size interval. The partition curve can be used to measure variability,
with the distance from the diagonal line indicating the degree of
unequivalence in the distribution of ensemble activity. The partition curve is
convex and lies below the diagonal (45 degree) line which represents
equivalence of normalized ensemble activity across time. The closer the convex
partition curve is to the diagonal line, the lower the ensemble activity
unequivalence. The further away the partition curve is from the 45 degree
diagonal line, the greater the unequivalence in the distribution of ensemble
activity. The partition curve shows that spike firing rates are not evenly
distributed and exhibit a large amount of variability in firing rate. It can
be calculated for nonstationary data (using a moving window) and used to
compare the distribution of the variable over time. The partition function is
a measure of the volume occupied by the system in phase space and can be
helpful for identifying trends in the distribution of the variable and
understanding how it is affected by changes in other variables. Color-coded
curves in Fig.5 show the partition curves for various scales (darker colors
represent finer scales while brighter colors represent coarser scales). With
increasing coarse-graining, the partition curves approach closer to the
diagonal and random partition curves but always maintain the characteristic
non-equivalence property. The results show that spike firing rates are not
evenly distributed and exhibit a large amount of variability in firing rate.
With increasing coarse-graining, the partition curves approach closer to the
diagonal and random partition curves but always maintain the characteristic
non-equivalence property. This transition represents how many microstates are
accessible to the system in a given ensemble under a given coarse-graining.
The black partition curve is calculated from a normal distribution for
comparison. This curve stand clearly separate from the partition curves of the
collected data, representing that the observed behavior is not a result of a
random process.

Figure 5:  Ensemble activity partition curve. In each panel, partition curves
for various scales are displayed against the diagonal (magenta) line of
equivalence. A perfectly equal ensemble size distribution would be one in
which every ensemble size has the same frequency (magenta line). The black
curve represents the partition curve for the normal distribution surrogate
series. Partition curves for different scales are color-coded, with brighter
colors indicating coarser scales and darker colors indicating finer scales.
With increasing coarse-graining, the partition curves approach closer to the
diagonal and random partition curves but always maintain the characteristic
non-equivalence property.

###  .3 Symmetry Breaking

####  .3.1 Symmetry breaking in wake/sleep states

As we move towards more coarse-grained scales, the joint distribution of
coarse-grained variables tends to become simpler. This is because macroscopic
behaviors, which are observed at coarser scales, are often simpler and more
universal compared to their microscopic counterparts. The “fixed point of the
renormalization group” refers to a state where a system remains unchanged (or
invariant) under the renormalization group transformation. This property of
invariance under transformations is considered a fundamental test for
criticality, and believed to hold significance beyond just thermodynamic
analogies [54]. Integrating out high-momentum degrees of freedom and rescaling
the remaining one can simplify the study of systems that exhibit behaviors on
many different length scales (as shown here, and previously [20, 54]).However,
simplicity at macroscale should not be confused with reducibility to purely
mechanistic microscopic phenomena.

In the context of cortical computation, major functional states arise through
interaction of numerous excitatory and inhibitory neurons. An understanding of
dynamics and information processing can be achieved through the study of
symmetry breaking, which can be understood as the deviation from a balanced
state of excitation and inhibition. Both renormalization and symmetry breaking
could potentially be observed and studied using multiscale co-occurrence
feature analysis of EIS (excitation/inhibition/scale) space. To analyze the
properties of the joint probability of the fraction of Excitation and
Inhibition across many temporal scales, we implemented a multidimensional
feature analyses. From the multidimensional space of
excitation/inhibition/scale (EIS), we calculated a “Multiscale Normalized Co-
occurrence Matrix” (_MNCM_) for multiple distances and orientations following
the volumetric adaptation [55, 56] of textural analysis [57]. MNCM was created
by calculating the frequency of occurrence of pairs of neighboring discrete
points (pixel) in the multidimensional EIS space. The spatial relationship
between the pixels can be defined using an “Offset” parameter (where the
offset relates to the spatial co-occurrence of E vs I in the EIS). Each entry
in the MNCM indicates the number of times a pixel with value $i$ is adjacent
to a pixel with value $j$. This spatial dependence matrix provides the
distribution of similar values (of E and I) at given offsets (i.e. the number
of times that a given value of ensemble E co-occurred with a given value of
ensemble I, considering the scale). We then used _MNCM_ to measure features of
distinct properties, such as contrast category (contrast, homogeneity,
dissimilarity), descriptive measures (mean, variance, correlation), and
orderliness features (maximum probability, entropy, angular second moment or
energy).

The multiscale co-occurrence matrix is analyzed using three groups of
measures: the Descriptive Statistics Group, the Contrast Group, and the
Orderliness Group. Each group captures different aspects of the matrix. The
Descriptive Statistics Group uses statistical metrics to describe the
distribution of point intensities in a given scale, including a measure of
correlation between specified pairs. The Contrast Group measures variation in
intensity within a scale, including homogeneity, contrast, and dissimilarity.
The Orderliness Group measures order or regularity within a scale, including
energy and entropy. These features are computed for each scale independently
and then combined into a single feature vector representing the entire
multidimensional space.

Figure 6:  Multidimensional features of Excitation and Inhibition co-
occurrence across different states. In each vertical band, a distinct feature
(extracted from the multidimensional co-occurrence matrix of
Excitation/Inhibition/Scale) is shown for various states; Awake, REM (rapid-
eye movement), LSL (light sleep) and SWS (slow-wave sleep) from left to right.
The features belong to three main categories: _contrast group_ (contrast,
homogeneity and dissimilarity), _descriptive group_ (correlation), and
_orderliness group_ (entropy and energy). Colors brighten as the distance of
the matched pairs in the co-occurrence matrix increases. The radius of each
circle shows the std. The features define the state-dependent properties of
Excitation and Inhibition interaction in a multiscale computational framework.
Note that each category of measures is normalized to the maximum across
different scales.

Statistical descriptions of these feature vectors show marked state-dependent
differences across these groups of measures (see Fig. 6). For example, the
multiscale energy of the system distinguishes REM from awake state, while
contrast is maximally distance-dependent across domains in light sleep (LSL)
and deep sleep (SWS). These calculations also show that homogeneity across
multiple scales is more prominent in wakefulness and less correlated with
distance in deep sleep. High contrast indicates a large amount of variability
or differences in the distribution of excitatory and inhibitory activity. If
contrast is distance-dependent across domains in light sleep and deep sleep,
this could suggest spatial symmetry breaking – different times of sleep (like
UP or DOWN states in SWS) may exhibit different balances of excitation and
inhibition, perhaps due to differentially tuned localized neuronal dynamics or
inputs (Note that distant-dependence here refer to the pair values in the
EIS). Energy and Entropy show opposing features of the wake-sleep cycle.
Energy, measure of uniformity or orderliness, is higher in awake or REM,
states that have more asynchronus activity. The distinction of REM from awake
state by the multiscale energy might reflect a state-dependent shift in the
balance of excitatory and inhibitory activity – a form of temporal symmetry
breaking - that is unique to information processing in dream versus
wakefulness. Entropy on the other hand is higher in SWS and LSL with REM
showing the least levels. This could be due to the various dissociation of
cortical events from thalamic events during non-awake states. Homogeneity and
Correlation showed less distinctive variations across different states.
Homogeneity, as a measure of the closeness of the distribution of elements in
the MNCM to the MNCM diagonal, increases with the uniformity of excitatory and
inhibitory activity. The prominence of homogeneity across multiple scales in
wakefulness and its lesser correlation with distance in light and deep sleep
may reflect a more detailed balance of excitation and inhibition during
wakefulness and more localized dynamics (LSL) yet more phasic undulation
during sleep. periods of intense neural activity and silence (UP and DOWN
states) that characterize non-rapid eye movement (NREM) sleep.

State-dependent differences in the balance of excitation and inhibition, and
thus the degree and nature of symmetry breaking, are suggested by the observed
differences across these measures. This could be due to changes in factors
such as neuronal firing rates, synaptic strengths, or neuromodulator levels
across different states of wake-sleep cycle. These distinctive multiscale
properties may result from intrinsic computational characteristics of
asynchronous irregular (AI), synchronous regular (SR), and asynchronous
regular (AR) regimes of stochastic network states [58, 59]. Further modeling
and theoretical studies are needed to clarify the links between stochastic
network states and multiscale features of excitatory-inhibitory balance
discussed here. The balance of excitation and inhibition in the system, and
hence the symmetry of the system, may be dynamically modulated across
different sleep states, possibly as a mechanism for achieving different
functional outcomes or maintaining homeostasis. The state-dependent changes in
multiscale co-occurrence features may reflect these underlying processes.
Further research, including a more detailed analysis of spatial and temporal
correlation with a higher number of recorded neurons, is needed to test and
refine this method.

####  .3.2 Symmetry Breaking and Balance Reset in Pathological States:
Insights from Seizures

Our recent study showed that the balance of excitation and inhibition breaks
down during electrographic seizures [20]. During a seizure, a shift in
excitatory and inhibitory firing can be observed (Figure 8A1,A2), along with
significant fluctuations in the oscillatory contents of the local field
potential (LFP), in addition to these spiking variations. We observed that the
average wavelet coherence displays an increase of high frequencies during a
seizure, followed by a return to lower frequencies. These changes in LFP
oscillations reflect previously identified frequency features of seizures (in
both electrocorticogram (ECOG) and LFP recordings) [60, 61, 62, 63]. LFP
comprises multiple oscillatory patterns simultaneously; simply focusing on the
wavelet modulus’s maxima doesn’t truly encapsulate the interaction of
oscillations and their alterations during seizure. Wavelet coherence is used
to establish the relation between the frequency structure of the LFP and the
event of interest [64]. The wavelet cross-spectrum is defined as the product
of the wavelet transform of a signal and the complex conjugate of the wavelet
transform of signal. Wavelet coherence is then defined as the squared absolute
value of the smoothed cross-spectrum normalized by the smoothed individual
power spectra from individual recording channels of the multielectrode
array.By applying a “multiridge” detection, strong components across multiple
concurrent frequency ranges of interest can be extracted. Normalizing the
average band ridges across frequency bands illustrates the abrupt frequency
shift around the seizure and indicates potential frequency changes over larger
time scales. The crazy-climber algorithm, a stochastic relaxation method based
on Markov chain Monte Carlo (MCMC), is used to extract ridges from the wavelet
[65, 66]. This method makes use of the energetic distribution in the time-
frequency representation of the LFP to extract multiple ridges across various
frequency ranges. Figure 7A shows these multiscale LFP coherence ridges. These
multiscale LFP coherence ridges show a sharp transition at the time that the
ensemble excitatory and inhibitory units undergo a major fluctuation,
indicating the onset of a seizure.

Figure 7:  Local field potential (LFP) multiscale during seizure. _A.
Multiscale coherence ridges._ Crazy-climber algorithm, a relaxation method
based on MCMC, was used to extract multiple ridges from the energetic
distribution in LFP wavelet coherence. Sharp transition in activity across
scales (frequency ranges) shows the emergence of seizure. _B. Multiscale
heatmap of synaptic current._ Synaptic current (estimated from the normalized
ensemble spiking) also shows a multiscale signature of pathological symmetry
breaking (during seizure) at the same time that multiridges of wavelet
coherence show a sharp transition from low to high frequencies.

Local field potentials (LFPs) are slow fluctuations of the extracellular
electric potential within brain tissue, arising from the combined electrical
activity of a large population of neurons in a localized region [67]. LFPs are
influenced by the synchronous input and output activity of many neurons,
including synaptic potentials and other slower transmembrane current flows
[68]. LFPs often reflect local information about processing in a given brain
region. However, the amplitude of the LFP is not a simple linear sum of
synaptic activity, as the extracellular space acts as a frequency-dependent
filter for the LFP. This means that the LFP is a filtered version of synaptic
activity, with higher frequency components attenuated more than lower ones
[69, 70, 71]. This has implications for interpreting LFP measurements, as the
LFP does not directly reflect raw synaptic activity. To address this issue, we
also examined the synaptic current estimated from the spiking neurons.

Synaptic currents are calculated by convolving individual spike trains with an
exponential kernel and normalizing based on the neuron count in each
functional category. The convolution process involves weighting each spike
with an exponential kernel, where weights decrease exponentially as the time
difference between the spike and the current time increases. After convolution
and normalization, the estimated synaptic current is obtained by subtracting
the ensemble of inhibition from the ensemble of excitation. This estimated
synaptic current represents a measure of the balance of excitation and
inhibition in the neuronal population, with positive values indicating a net
excitatory influence and negative values indicating a net inhibitory
influence. A heatmap of synaptic current, estimated from normalized ensemble
spiking, reveals a multiscale signature of pathological symmetry breaking
during seizure (Fig.7B). Two distinct bands of maximum difference,
corresponding to frequency perturbations observed in LFP oscillations ( 7A),
delineate the temporal zone of the seizure. The first band represents a clear
breakdown of balance, followed by another significant perturbation, possibly
signifying an attempt to restore order.

Figure 8:  Symmetry breaking in pathological state (seizure). _A1._ Rasterplot
of excitatory (blue) and inhibitory (red) unit activity during a 9 minute
epoch, _A2._ zoom in to the middle 80 sec. _B. Reset of balanced activity
after seizure._ Normalized cumulative activity of the signed ensembles (in
green) shows a deviation from the estimated balanced trajectory (red) at the
onset of seizure. Dark and light gray lines show Monte Carlo simulation of
trend stationary and difference stationary processes. The normalized
cumulative activity shows a return to the balance trajectory toward the end of
the epoch.

The breakdown of balance during seizure is followed by another interesting
phenomenon. A rasterplot of excitatory and inhibitory unit activity during a
540-second epoch shows the multiscale features of the imbalance leading to a
seizure (Fig.8A1). Multiscale refers to changes and effects observable at
various time (fine to coarse-grained scales) or spatial scales (spiking and
LFP). The functional restoration after a seizure points to the reset of
multiscale balance, as shown by the return of normalized cumulative activity
to the balance trajectory towards the end of the epoch. The extent of this
imbalance is marked by the deviation of the actual behavior (red line) from a
projected balanced path (green line). This reset is compared with simulated
timeseries representing stochastic (difference-stationary process) and
deterministic (trend-stationary) processes, neither of which shows a return to
the balanced state as observed in the real data. These results suggest that
the brain has multiscale adaptive features that help it return to a balanced
state after a seizure. This behavior was observed in N=6 seizure recordings
that we had from these patients. Though technically challenging, further
studies with many more seizures are needed to verify these properties. Whether
the balance breakdown and reset extends to other types of seizure (such as
non-local) is an interesting question that needs future testing.

## Discussion

The macroscopic behaviors of a system, which are observed at larger scales,
can often be described by simpler and more universal principles than the
microscopic mechanisms that govern the behavior of individual components of
the system [72]. Neural field models and neural mass models are different ways
to model how neurons act together. Neural field models account for the spatial
distribution of neuron groups in a coarse-grained area [73, 74] and explain
how a variable that captures neural activity changes over both space and time
[75]. Neural mass models only track activity over time [76, 77] and do not
account for spatial extensions. Both of these field models (i.e. neural mass
model and neural field model) use the same principles of population coding and
time-graining [78, 79]. Mean-field models have limitations in understanding
critical dynamics because they don’t account for the interaction between
degrees of freedom at different length scales. These models assume
fluctuations are Gaussian and are limited to studying the linear response to
stimuli. They neglect nonlinear interactions, which are essential for
computation at macroscopic scales. To fully understand brain dynamics and its
various types of criticality, more advanced methods beyond mean-field
approaches are necessary [80]. A solution is to use renormalization group that
works by repeatedly integrating out high-momentum degrees of freedom and
rescaling the remaining degrees of freedom [81, 82, 83]. In this context,
“degrees of freedom” refers to the number of independent ways that the
components of a system can change, while “high-momentum degrees of freedom”
often refer to the fast, microscopic details of the system, and “Rescaling”
means adjusting the system’s description to focus on larger-scale or slower
aspects. The significance of the invariance of probability distributions under
repeated coarse-graining, also known as a fixed point of the renormalization
group, is considered a fundamental test for criticality [54]. It suggests that
when a system exhibits this invariance, it may be in a critical state,
indicating its broader applicability of RG and relevance in understanding
complex systems. For example, phenomenological renormalization group (PRG) can
used to analyze single-neuron recordings (from a mouse’s hippocampus),
revealing some scaling characteristics remain in the supercritical regime for
a finite system [84].

In this study, we scrutinized the ensemble excitation and inhibition using a
renormalization group transformation. This process involved eliminating high-
momentum degrees of freedom, followed by rescaling the remaining degrees of
freedom. Subsequently, we conducted multiple tests to assess the invariance
properties of the system. Our findings indicate a consistent decrease in the
variance of ensemble Excitatory-Inhibitory (E-I) as we move from finer to
coarser scales, highlighting scale-invariance in the correlated fluctuations
of excitation and inhibition in the cerebral cortex. During the sleep-wake
cycle, a balance between excitatory and inhibitory activity was observed and
the balance was preserved across different scales and states. The EI balance,
integral to cortical computation, remains intact even as the distribution
narrows with coarser temporal scales. Furthermore, past studies have shown
that the timing of spikes is delicately adjusted by dynamic interactions
between excitatory and inhibitory conductances [29, 85]. The balance between
excitatory and inhibitory conductance helps to regulate the timing of the
action potentials [30, 86]. This fine-tuning is crucial for all sorts of
neural processes. Some simulations and experimental studies suggest that the
spatial distribution of inhibitory synaptic activity on pyramidal cells
predominantly controls the activity of the system, maintaining it in an
excitable state, poised for the next wave of thalamic excitation [31, 87]. So
in essence, the balance of excitatory and inhibitory influences leaves the
cortex in a state where it’s ready to respond quickly and appropriately to new
information coming in, indicating that the balance is a key feature of
cortical computation. It’s important to note that there is a wide parameter
space where balance can be maintained while computational processing of inputs
occurs [28, 88].

We utilized collapse curve analysis to examine the multiscale attributes of
cortical ensemble excitation-inhibition balance fluctuations throughout
various wake-sleep cycle stages, offering insights into the system’s self-
similarity or universality. Significantly, various randomization methods
resulted in surrogates that displayed similar behavior and didn’t collapse
onto the data curve, underscoring that the observed balance isn’t simply a
random statistical characteristic, but instead is connected to a collective
ensemble spiking pattern. As the analysis reached the coarsest scale, the
surrogate collapse curve started to align with the data collapse curve,
implying that although the balance remains, the packet content might forfeit
significant structure from an information processing standpoint at very
extended timescales. Moreover, different states appeared to collapse onto the
same regime, highlighting a consistent balance across scales. This exploration
of scale-invariant behavior within cortical networks provides a fresh
viewpoint on criticality, proposing that the system exhibits this behavior
across a vast range of parameters due to potential competing interactions or
feedback mechanisms in cortical networks, as opposed to at a single critical
point. As we have suggested before, this could be due to multiple interacting
factors or feedback mechanisms in the cortical networks [38, 21].

We also developed “partition curve” method for analyzing the distribution of
ensemble activity. The partition curve offers a way to assess ensemble
activity distribution based on size or intervals. Notably, the curve can shed
light on the level of equality, or lack thereof, in ensemble activity
distribution. When used in conjunction with Hawkes theory, this tool could aid
in the exploration of neuron ensemble spiking patterns. A noteworthy finding
from our partition curve was the significant variability in spike firing
rates, suggesting a non-uniform distribution. In the case of nonstationary
data, the partition curve can monitor temporal changes. Further analysis
comparing our findings with a normal distribution supported the conclusion
that the patterns observed were not simply the result of random processes.
This implies that the patterns and distributions identified in the neuronal
ensemble activity are potentially meaningful and could provide valuable
insights into how these neurons function as a group. At coarser scales, the
partition curves appeared to converge towards the diagonal and random
partition curves, whilst preserving a unique non-equivalence attribute. This
shift signifies the spectrum of microstates - or short-term, specific states
of activity - that the system can access within an ensemble at distinct
coarse-graining levels. In statistical mechanics, a degree of freedom is a
single scalar number describing the microstate of a system. The specification
of all microstates of a system is a point in the system’s phase space.
Entropy, in both thermodynamics and information theory, is a measure of the
number of possible microstates or configurations that a system can have. In
this sense, there is a connection between degrees of freedom, states of the
system, and information (in terms of entropy). The more degrees of freedom a
system has, the more possible microstates it can have, and the higher its
entropy will be. This means that system at finer scales, with more degrees of
freedom, can contain more information.

We also introduce multiscale descriptors that examine degrees of freedom,
rescaling, and the simplification of joint distributions under coarse-
graining, particularly in the context of criticality and macroscopic behavior.
These features, computed from a Multiscale Normalized Co-occurrence Matrix
(MNCM), are descriptive statistics that can capture different aspects of the
spatial distribution of intensities in the multidimensional landscape of
excitation/inhibition/scale. Specific attributes, like correlation, contrast,
homogeneity, energy, and entropy, offer unique insights into the system’s
behavior. This reduces the complexity of the multiscale data to a set of
numbers that describe key aspects of its structure. The MNCM analysis in this
study is applied across multiple temporal scales to shine light on the
collective behavior across scales, effectively quantifying “the degrees of
freedom” of the system as it undergoes rescaling. Specifically, multiscale
energy, indicating uniformity, differentiates REM sleep from wakefulness,
suggesting unique shifts in neural activity balance. Contrast, which measures
variability, exhibits a strong distance-dependency in light and deep sleep,
hinting at spatial symmetry breaking. Homogeneity, indicating uniform
activity, is pronounced in wakefulness and less distance-correlated in deep
sleep, indicating intricate neural balances and localized dynamics,
respectively. Entropy, measuring disorder, is highest in deep and light sleep
and lowest in REM, reflecting the dissociation of cortical and thalamic events
in non-awake states. In biological neuronal networks, symmetry breaking might
be seen in the transition from balanced, coordinated activity to a state where
certain neural pathways are favored over others. The MNCM could potentially
capture such transitions by quantifying the joint probabilities of different
patterns of excitation and inhibition and discern state-dependent differences
in neural activity.

###  .4 Concluding Remarks

In his seminal paper _More is different_ , Philip Anderson speculated that
“internal structure of a physical matter does not need to be symmetrical even
if total state of it is” and “the state of a really big system does not have
to have the symmetry of the laws which governs it” [89]. He argues that a key
feature in biological systems may be the existence of temporal ordering and
periodicity in addition to their broken symmetry. We suggest that the nervous
system resides at a unique position in the abstract computational state space:
while the overall ensemble excitatory-inhibitory balance is both scale-
invariant and state-independent (wakefulness, REM, or NREM), at any given time
t, the momentary dominance of either excitatory or inhibitory neurons is more
pronounced at finer time scales and the degree to which this happens is state-
dependent. As we move towards coarser time scales, the system becomes more
rigid with respect to the possibility of momentary symmetry breaking. Symmetry
breaking is the bedrock of information [90, 91] and momentary symmetry
breaking in cortex is utilized to process information (awake), consolidate
memory (slow-wave sleep), or create new associations (REM). However, these
symmetry-breaking events with dominance of excitatory or inhibitory neurons
are then compensated by opposite dominance, maintaining a scale-invariant
overall balance of ensemble excitatory-inhibitory neurons. In a sense, it
seems that neocortex does not have the rigid symmetries and end-state
spontaneous symmetry breaking of crystals or magnets that bring those systems
to a minimum energy state. Rather, to maintain the capacity to process
information reliably, cortex sacrifices the minimum energy state through a
rebound after any symmetry breaking.

The dynamic properties of ensemble excitatory-inhibitory balance show
parallels with symmetry breaking in dissipative structures [92, 93].
Previously, the link between the nature of dissipative non-equilibrium and
symmetries in dynamical systems prompted considerations of symmetry breaking
in large-scale neural activity [94]. This heuristic model of visual
hallucination (an extension of [95]) has certain biologically implausible
features (such as homogeneous planar sheet of neurons with radial isotropic
connections), but the presence of both excitatory and inhibitory neurons and
recurrent feedback enables the neural field equation to produce a variety of
spatial patterns when instability of the homogeneous resting state arises.
What is shared between this model, our quantitative frame, and dissipative
structures is the presence of symmetry breaking and unstable symmetry as their
dynamical signatures.

In fact, the instability of symmetry in dissipative structures may be the
reason why biological systems need to constantly harvest energy to return to
the unstable symmetric state [96]. We speculate that perhaps the significant
energy usage in cortex is to re-establish the unstable symmetry that is
constantly breaking during information processing. Dissipative structures
share many attributes with biological systems, such as multistability,
sustained oscillations, and spatiotemporal patterns manifesting as propagating
waves [97]. Our prior examinations of cortical field potentials show the
presence of anisotropic traveling waves and dominant oscillatory events across
the wake/sleep cycle [36]. These are signatures of a complex adaptive system
with many computational length scales, lending itself to a form of invariant
ensemble excitatory-inhibitory balance where momentary symmetry breaking at
fine timescales provides the capacity to process information.

Note that seizures are manifested by abnormal symmetry breaking that does not
stay limited to momentary excitatory or inhibitory dominance but rather turns
to disruption of ensemble balance across all scales. In parallel with this
multiscale breakdown in ensemble excitatory-inhibitory balance, a shift of
spiking energy in the estimated synaptic current coincides with transient
oscillatory changes in LFP (a rapid $\delta$-like rise of higher frequencies
and suppression of slow frequencies). During this period of abnormal symmetry
breaking, information processing comes to a halt and the patient suffers from
loss of consciousness. The multiscale breakdown of balance is then followed by
a return to a balanced state, further confirming the presence of scale-
invariant balance in a complex adaptive fashion.

It is important to emphasize the distinction between the simplicity of laws
and their outcomes. The outcomes, such as the intricate and seemingly chaotic
patterns of spiking activity, do not have to possess the same simplicity as
the underlying organizing principles, namely the scale-invariant ensemble
excitatory-inhibitory balance. By quantifying the catalog of invariances
(balance across states) at different temporal and spatial scales, we can
search for governing principles and diverse neural activity patterns that
emerge from these simple laws.

## Methods

###  .5 Experimental data

For this study, we used recordings from multielectrode array implants in
layers II/III of the human temporal cortex. The electrodes in these arrays are
arranged in a 10x10 grid with $400\mu m$ spacing, covering an area of 4x4
millimeters (see Fig.1). The four corner electrodes are used for grounding,
leaving a total of 96 active recording electrodes. Data is initially sampled
at 30kHz, then filtered and thresholded for spike detection. We used a
combination of spike-waveform characteristics and the sign of the short-
latency cross-correlogram of spike time pairs to classify units as either
excitatory (E) or inhibitory (I). Video, electrocorticogram (ECOG), and scalp
EEG were used to classify the 12-hour recordings into state labels: Awake,
Light Sleep (stages II and III), Deep Sleep (Slow-Wave Sleep), and Rapid Eye
Movement (REM). For further details on recording methods and morpho-functional
characterization of excitatory and inhibitory neurons, see the cited
references [98, 99].

###  .6 Renormalization: Spiking coarse-graining and field estimates

####  .6.1 Ensemble Excitation and Inhibition.

If we define each neuron as an independent particle and its stochastic spiking
as random fluctuations in space, the observed randomness can be absorbed by
redefining a cluster of neurons (instead of a single neuron) to account for
self-interactions, pairwise interactions, and higher-order interactions. This
ensemble is then temporally coarse-grained by summing all spikes in a given
time-bin, with scales ranging from 1 millisecond to 10 seconds (equally spaced
in a logarithmic fashion) to specify the scale of observation. The logarithmic
scale is chosen to reduce computational burden, with a denser spread of scales
at finer time resolutions.

While these recordings yield a high number of neurons from each array, we face
both massive subsampling (in our case, hundreds of neurons out of tens of
thousands in the columnar region of recording) and spatial nonuniformity.
Although we had a 4/1 match of the number of excitatory and inhibitory neurons
with anatomical data and a 4/1 higher spiking rate of inhibitory to excitatory
neurons [100, 98], given the subsampling and nonuniformity, the ensemble
series are normalized by the number of neurons. This procedure was done
independently for excitatory and inhibitory cell categories to yield ensemble
fractions of excitation and inhibition.

####  .6.2 Synaptic current.

To explain the multiscale dynamics of spiking excitation and inhibition with
multiscale representation of local field potentials (LFPs) using their wavelet
transforms, we used unit activity to reconstruct estimated synaptic current.
It is suggested that synaptic current generates LFPs [101, 102], which reflect
a field measure of the tens of thousands of neurons present in the recording
area from which spikes are sampled. The modeling involved convolving each
spike with an exponential kernel [103].

$C(t)=\int_{-\infty}^{\infty}D({t}^{\prime})exp[-(t-{t}^{\prime}/\tau_{s})]d{t}^{\prime}$
(1)

Here:

$C(t)$ is the estimated synaptic current, $D(t)$ represents the spike trains,
$\tau_{s}$ is the time constant for the synaptic current.

Different time constants ($\tau_{s}$) are used for the convolution depending
on whether the spike is excitatory (3 milliseconds) or inhibitory (10
milliseconds). This is based on prior studies that found differences in the
time constants for excitatory and inhibitory synaptic currents [30]. Synaptic
current at any time is a weighted sum (or integral, in the continuous case) of
the spike train, where the weights decrease exponentially as the time
difference between the spike and the current time increases. After convolving
the spike trains with their respective kernels, the estimated synaptic current
is obtained by subtracting the ensemble of inhibition from the ensemble of
excitation. This result is then corrected by the number of cells in each
category (excitatory or inhibitory).

###  .7 Randomization of ensemble activity

####  .7.1 Random permutation.

We used this test to verify that randomizing the aggregate spike series by
itself cannot mimic the observed excitatory-inhibitory balance. After
calculating the inter-spike interval (ISI) for the pooled ensemble series of
in a given functional category (Excitatory or Inhibitory), a random
permutation of the ISI was performed for that functional category ensemble
series. This was followed by a cumulative summation of the ISI to create a new
temporal order of ensemble spikes. This process ensures that the randomized
ensemble series has the same number of spikes and set of ISIs as the original
ensemble series, but with a different arrangement of spikes within the series.
The observations verify that the excitatory and inhibitory balance is not
reproducible by randomization of ensemble spikes and the observed fluctuations
$f_{obs}=Ens_{E}-Ens_{I}$ are not results of random events such as
$f_{randperm}=SurrogateEns_{E}-SurrogateEns_{I}$.

####  .7.2 Fixed-ISI circular shift of spike ensemble.

Before creating the ensemble series, the inter-spike interval (ISI) of each
unit’s spike series was calculated (by “unit” here, we refer to a single
neuron). The spikes of each unit were then shifted based on a random value
between 1 and the maximum ISI of that unit’s spike series. The randomized
units were then aggregated to create the randomized ensemble series of a given
functional category (Excitation or Inhibition). This process ensures that the
resulting ensemble series is made up of units with intact internal spike
timing but with disrupted timing between units in that functional category.
The results show that by destroying the relation between ensemble inhibition
and ensemble excitation while preserving their internal structure, the
observed fluctuations and most importantly, the tightly bound relation of
Excitation and Inhibition is lost.

###  .8 Multiscale collapse of ensemble activity

The observed fluctuations of ensemble activity, represented by
$f_{obs}=Ens_{E}-Ens_{I}$, were first sorted in ascending order. A cumulative
summation was then performed on the series resulting in a parabolic curve
showing mirror symmetry due to the overall balance of ensemble Excitation and
Inhibition (Fig.4). To examine self-similar or self-affine characteristics, a
collapse curve was obtained by rescaling the curve for a given scale so that
it was bounded between 0 and 1 (x-axis normalization). We then systematically
identified non-collapsible regions for each of the comparisons (A: data
against random series; B: comparison of different scales; C: comparison of
different states across various scales). A collapse curve is then a rescaled
shape (of cumulative sum of sorted fluctuation of ensemble activity) for
different duration that collapses onto a single universal curve.

The general form of collapse curve for ensemble excitation and inhibition can
be written as:

$C(t)=\frac{F(t)-F_{min}}{F_{max}-F_{min}}$ (2)

, where $C(t)$ is the collapse curve, $F(t)$ is the cumulative sum of sorted
fluctuation of ensemble activity, $F_{min}$ and $F_{max}$ are the minimum and
maximum values of $F(t)$, respectively, and $t$ is the time scale.

###  .9 Ensemble activity partition

Let $A$ denote the fraction of ensemble activity of a given functional
category (either excitation or inhibition), let $B$ denote the discrete order
of $A$ given by values $b_{1},b_{2}...,b_{n}$ (in non-decreasing order) and
their probabilities $g(b_{j}):=\Pr(B=b_{j})$. The partition curve is defined
as a continuous piece-wise linear function $(P)$ connecting the points
$(C_{i},P_{i})$ such that:

$\displaystyle C_{i}$ $\displaystyle:=\sum_{j=1}^{i}g(b_{j})$ (3)
$\displaystyle D_{i}$ $\displaystyle:=\sum_{j=1}^{i}g(b_{j})\,b_{j}$
$\displaystyle P_{i}$ $\displaystyle:={\frac{D_{i}}{D_{n}}}$

,with $B$ bounded between 0 and 1, the partition curve represents a size-based
distribution (as a list of values) that indicates the percentage of ensemble
activity of a certain size or in a certain size interval. The partition curve
is convex and lies below the diagonal (45 degree) line which represents
equivalence of normalized ensemble activity across time. The closer the convex
partition curve is to the diagonal line, the lower the ensemble activity
unequivalence. The further away the partition curve is from the 45 degree
diagonal line, the greater the unequivalence in the distribution of ensemble
activity. Color-coded curves in Fig.5 show the partition curves for various
scales (darker colors represent finer scales while brighter colors represent
coarser scales). The black partition curve is calculated from a normal
distribution for comparison.

###  .10 Multiscale descriptors of co-occurrence of Excitation and Inhibition

To analyze the properties of the joint probability of the fraction of
Excitation and Inhibition across many temporal scales, we implemented a
multidimensional textural feature analyses. From the multidimensional space of
excitation/inhibition/scale (EIS), we calculated a “Multiscale Normalized Co-
occurrence Matrix” (_MNCM_) for multiple distances and orientations following
the volumetric adaptation [55, 56] of textural analysis [57]. MNCM was created
by calculating the frequency of occurrence of pairs of neighboring discrete
points (pixel) in the multidimensional EIS space with specific normalized
values. The spatial relationship between the pixels can be defined using an
“offset” parameter (where the offset relates to the spatial co-occurrence of E
vs I in the EIS). Each entry in the M-CM indicates the number of times a pixel
with value $i$ is adjacent to a pixel with value $j$. This spatial dependence
matrix provides the distribution of similar values (of E and I) at given
offsets (i.e. the number of times that a given value of ensemble E co-occurred
with a given value of ensemble I, considering the EIS). We then used _MNCM_ to
measure features of distinct properties, such as contrast category (contrast,
homogeneity, dissimilarity), descriptive measures (mean, variance,
correlation), and orderliness features (entropy, angular second moment or
energy).

Considering $P(i,j)$ as the joint distribution of $i$ (ensemble excitation
dimension) and $j$ (ensemble inhibition dimension), of the original
multidimensional matrix, then $p(i,j)$ is the $(i,j)th$ entry in a normalized
MNCM (i.e. spatial dependence matrix). The normalization of $P(i,j)$ is
performed using the sum of all its values. $N_{g}$ is the number of distinct
normalized levels in the multidimensional space of excitation/inhibition/scale
obtained after the operation of _MNCM_.

The features from the category of “descriptive statistics” were defined as:

$Correlation=\cfrac{\sum_{i}\sum_{j}(i,j)p(i,j)-u_{x}u_{y}}{\sigma_{x}\sigma_{y}},\\\
$ (4)

where $px$ and $py$ are the partial probability density functions; $u$ and
$\sigma$ are their mean and std, respectively.

Measures from the “contrast” group were:

$Contrast=\sum_{i=1}^{N_{g}}\sum_{j=1}^{N_{g}}(i-j)^{2}p(i,j)\\\ $ (5)

$Dissimilarity=\sum_{i=1}^{N_{g}}\sum_{j=1}^{N_{g}}(i-j)p(i,j)\\\ $ (6)

(similar to contrast, but instead, with linear weights.)

$Homogeneity=\sum_{i=1}^{N_{g}-1}\sum_{i=1}^{N_{g}-1}\cfrac{1}{1+(i-j)^{2}}p(i,j)$
(7)

and the features of “orderliness” were:

$Energy=\sum_{i=1}^{N_{g}}\sum_{j=1}^{N_{g}}p(i,j)^{2}\equiv
sqrt(\>Angular\>Second\>Moment)\\\ $ (8)

$Entropy=-\sum_{i=1}^{N_{g}}\sum_{j=1}^{N_{g}}p(i,j)log(p(i,j))$ (9)

The “Descriptive Statistics Group”, “Contrast Group”, and “Orderliness Group”
capture different aspects of the MNCM: ‘ _Descriptive Statistics Group_ is
characterized by statistical metrics that describe the overall distribution of
point intensities in a given scale. One key feature of this group is
“Correlation”, which measures the joint probability occurrence of specified
pairs. _Contrast Group_ includes several measures of contrast or variation in
intensity within an scale. “Homogeneity” measures the closeness of the
distribution of elements in the MNCM to the MNCM diagonal. “Contrast” measures
the local variations in the MNCM. “Dissimilarity” measures the variation of
pairs in the MNCM. _Orderliness Group_ includes measures of order or
regularity within a given scale. “Energy” (also known as second angular
moment) provides the sum of squared elements in the MNCM, and “Entropy”
measures the complexity or disorder in a given scale. The application to
volumetric (multiscale) data involves computing these features for each scale
independently and then combine the resulting feature vectors into a single
feature vector representing the entire multidimensional space.

###  .11 Wavelet Coherence and Ridge Detection

To relate the frequency structure of the local field potential (LFP) with the
estimated synaptic current, we used wavelets to calculate coherence [64]. The
wavelet cross-spectrum is defined as the product of the wavelet transform of
signal $(x)$ and the complex conjugate of the wavelet transform of signal
$(y)$. The squared absolute value of the smoothed cross-spectrum normalized by
the smoothed individual power spectra defines wavelet coherence.

$W_{n}^{XY}(s)=W_{n}^{X}(s)W_{n}^{Y^{*}}(s)\\\ $ (10)
$C_{n}^{2}(s)=\frac{\left|\left\langle
W_{n}^{XY}(s)\right\rangle.s^{-1}\right|^{2}}{\left\langle\left|W_{n}^{XX}(s)\right|.s^{-1}\right\rangle\left\langle\left|W_{n}^{YY}(s)\right|.s^{-1}\right\rangle}$
(11)

To extract ridges from the wavelet, we used the crazy-climber algorithm. This
stochastic relaxation method is based on Markov chain Monte Carlo (MCMC) and
can extract multiple ridges from analyzed signals [65, 66]. In its algorithmic
procedure, the energetic distribution in the time-frequency representation of
the LFP is used to extract multiple ridges across frequency ranges of LFP.

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