Imaginary Part of Coherence

Measuring how distinct brain regions communicate is a cornerstone of modern neuroscience, offering insights into everything from simple reflexes to complex cognition. However, a significant obstacle stands in the way: the very physics of how we record brain activity. Techniques like EEG and MEG are plagued by artifacts where signals from a single source spread across multiple sensors, creating illusory connections that can mislead researchers. This article tackles this fundamental problem of "ghost" correlations, which appear real but are merely echoes within the brain's conductive medium.

In the chapters that follow, we will unravel this issue and its elegant solution. The journey begins with the "Principles and Mechanisms," where we will delve into the mathematics of complex coherence to understand how it can distinguish real, time-delayed interactions from instantaneous artifacts. We will then broaden our perspective in "Applications and Interdisciplinary Connections," exploring how this powerful method provides a practical filter for neuroscientists and, surprisingly, how the same core principle of separating real and imaginary components appears in the disparate field of quantum physics.

Imagine you are at a concert, trying to understand the beautiful interplay between a violin and a cello. You have two microphones, but they are not perfect. The microphone pointed at the violin also picks up a faint echo of the cello, and vice versa. How can you be sure that a pattern you hear is a true duet—a conversation between the two instruments—and not just a "ghost" created by your imperfect recording? This is the very conundrum neuroscientists face when they listen to the symphony of the brain.

When we place sensors on the scalp (Electroencephalography, or EEG) or just outside it (Magnetoencephalography, or MEG), we are eavesdropping on the electrical activity of millions of neurons. These signals travel through the brain tissue, cerebrospinal fluid, skull, and scalp to reach our sensors. This journey is not a clean one. The electrical fields spread out, a phenomenon known as ​​volume conduction​​ or ​​field spread​​. Just like the microphones at the concert, a single sensor doesn't just record the activity from the brain region directly beneath it; it picks up a mixture of signals from many different, often distant, sources.

This mixing poses a fundamental problem. Suppose we are looking at the signals from two sensors, $x(t)$ and $y(t)$. We might see that they oscillate in perfect lockstep. A natural first guess would be that the two brain regions they are measuring are communicating directly and instantly. But it's far more likely that both sensors are simply "listening" to the same powerful, underlying source—a "common source" that has been smeared across the brain by volume conduction. This creates a spurious, or fake, correlation. It looks like a connection, but it's just an echo.

The key property of this echo is that it is, for all practical purposes, ​​instantaneous​​. There is no meaningful time delay for the signal from a single source to arrive at two different nearby sensors. This zero-lag nature is the artifact's Achilles' heel, and exploiting it is the secret to telling the echo apart from the true conversation.

To see how this works, we need to change our perspective. Instead of looking at the wiggles of our signals $x(t)$ and $y(t)$ in time, we can look at them through "frequency goggles." Using a mathematical tool called the Fourier transform, we can break down any signal into the different rhythms, or frequencies, of which it is composed.

To measure the relationship between two signals, $x(t)$ and $y(t)$, at a specific frequency $f$, we use a quantity called the ​​cross-spectrum​​, denoted $S_{xy}(f)$. Now, here is the beautiful part. The cross-spectrum is not just a number that tells you "how much" the signals are related. It is a ​​complex number​​.

Don't let the word "complex" scare you. A complex number is simply a number with two parts: a ​​real part​​ and an ​​imaginary part​​. You can think of it as a point on a 2D map. The real part tells you how far you are on the east-west axis, and the imaginary part tells you how far you are on the north-south axis. Or, even better, you can describe this point by its distance from the origin (its magnitude) and its angle (its phase).

For the cross-spectrum $S_{xy}(f)$, these two parts have profound physical meaning:

• The ​​magnitude​​, $|S_{xy}(f)|$, tells us about the strength of the shared oscillations between $x(t)$ and $y(t)$ at frequency $f$.
• The ​​phase​​, $\phi(f)$, tells us the average time delay between the two signals' rhythms.

A time delay, $\tau$, between two signals translates directly into a phase lag, $\phi$, at a given frequency $f$ by the simple relation $\phi = 2\pi f \tau$.

Now we have the key to unmasking our artifact. Volume conduction is an instantaneous mixing phenomenon. The time delay $\tau$ is zero. If we plug $\tau = 0$ into our phase equation, we get $\phi = 2\pi f \cdot 0 = 0$. A phase of zero!

What does a complex number with a phase of zero look like on our 2D map? It lies purely on the horizontal, east-west axis. It has no north-south component. In other words, its imaginary part is zero.

This is a stunningly simple and powerful result. Any relationship between two signals that is caused by instantaneous mixing—be it from volume conduction, field spread, or even a shared recording reference—will contribute only to the real part of the cross-spectrum. We can prove this quite elegantly. In a linear mixing model with a real-valued mixing matrix $A$, the sensor cross-spectral matrix $S_x(f)$ is related to the source cross-spectral matrix $S_s(f)$ by $S_x(f) = A S_s(f) A^T$. If the sources are uncorrelated, their cross-spectral matrix $S_s(f)$ is a real, diagonal matrix. Since $A$ is also real, the resulting sensor cross-spectral matrix $S_x(f)$ must be purely real. Its imaginary part is identically zero.

The strategy to defeat the artifact is now wonderfully obvious: just ignore the real part! We can design a connectivity measure that is sensitive only to the imaginary part of the cross-spectrum. This measure is fittingly called the ​​imaginary part of coherence​​ (often abbreviated as ImCoh or iCOH).

Coherence is simply the cross-spectrum normalized by the power of the individual signals, to give a value between 0 and 1. The complex coherency is:

Since the denominator is a real number, the imaginary part of the coherency is simply:

This measure, by its very construction, is blind to any zero-lag effects. It acts as a perfect filter, automatically rejecting the spurious connections generated by volume conduction while remaining sensitive to true, time-lagged interactions.

A genuine interaction between two brain areas involves signals propagating along axons and crossing synapses. This process takes time. For a pure time delay $\tau$, the coherency between the signals becomes $C_{xy}(f) = \exp(i 2\pi f \tau) = \cos(2\pi f \tau) + i \sin(2\pi f \tau)$. The imaginary part is $\sin(2\pi f \tau)$. This is only zero if the delay $\tau$ is zero. Any real biological interaction with a non-zero delay will generate a non-zero imaginary coherence.

This all sounds wonderful in theory, but does it hold up in practice? Let's consider a real-world scenario.

Suppose we are recording from two cortical areas, A and B, that are about $L=30\,\mathrm{mm}$ apart. We know from biology that signals in myelinated cortico-cortical axons travel at roughly $v=3\,\mathrm{m/s}$. The expected travel time is therefore $\tau = L/v = (30 \times 10^{-3}\,\mathrm{m}) / (3\,\mathrm{m/s}) = 0.01\,\mathrm{s}$, or $10\,\mathrm{ms}$.

If we are looking at brain rhythms in the alpha band, say at $f=10\,\mathrm{Hz}$, this delay should produce a phase lag of $\phi_{\text{exp}} = 2\pi f \tau = 2\pi \times 10 \times 0.01 = 0.2\pi$ radians (or 36 degrees).

Now, let's say our measurement gives us a cross-spectrum with a real part of $R=3\,\mu\mathrm{V}^2/\mathrm{Hz}$ and an imaginary part of $I=3\,\mu\mathrm{V}^2/\mathrm{Hz}$. The observed phase lag is $\phi_{\text{obs}} = \arctan(I/R) = \arctan(1) = \pi/4$ radians (or 45 degrees). This corresponds to a total delay of $12.5\,\mathrm{ms}$.

Look how close these values are! The observed delay ($12.5\,\mathrm{ms}$) is very similar to the delay we predicted from basic anatomy and physiology ($10\,\mathrm{ms}$). The small difference of $2.5\,\mathrm{ms}$ is perfectly plausible, as it could account for other delays like synaptic transmission. This gives us confidence that the non-zero imaginary coherence we measured is not just noise, but a reflection of a genuine, physiological connection between the two brain regions.

The principle of rejecting zero-lag contributions is so powerful that several other connectivity metrics have been built upon it. The ​​Phase Lag Index (PLI)​​ and ​​Weighted Phase Lag Index (wPLI)​​ are cousins of imaginary coherence. Instead of using the cross-spectrum, they look only at the distribution of phase differences between signals, but the core idea is the same: they are designed to be zero if the phase differences are symmetrically distributed around zero, as would be the case for volume conduction.

It's also useful to contrast these methods with others, like the ​​Phase-Locking Value (PLV)​​. The PLV measures the consistency of the phase relationship, but it doesn't care what that phase is. A perfect zero-lag relationship and a perfect non-zero-lag relationship both give a PLV of 1. Therefore, PLV is sensitive to volume conduction artifacts.

But is there a catch? By throwing away the real part of the cross-spectrum, are we losing anything important? Yes. We are making a fundamental trade-off. It is conceivable that some forms of neural communication, perhaps mediated by electrical synapses (gap junctions), are truly instantaneous. If such a genuine zero-lag interaction exists, imaginary coherence will be completely blind to it. We accept this potential "false negative" in order to gain robustness against the much more common and insidious artifact of volume conduction. We choose to be conservative, ensuring that the connections we report are highly likely to be real, at the cost of possibly missing a specific class of true interactions.

This is the beauty of the scientific method in action. We are faced with a contaminated measurement, but by understanding the physical nature of the artifact—its instantaneousness—we can devise a clever mathematical strategy to look right past it. The imaginary part of coherence provides a stunningly elegant lens, allowing us to filter out the echoes and listen to the true, time-delayed conversations that form the symphony of the human brain.

Having journeyed through the principles of complex coherence, we might be tempted to view it as an elegant but abstract mathematical construct. Nothing could be further from the truth. Like a master key that unexpectedly unlocks doors in rooms we never thought were connected, the concept of coherence—and particularly its imaginary part—proves to be an indispensable tool for discovery across a surprising range of scientific disciplines. It allows us to solve very practical puzzles, from untangling the whispers of neurons in the living brain to understanding how a single atom drinks in a beam of light.

Let's begin in the world of neuroscience. One of the grandest challenges is to map the brain's "wiring diagram" not just anatomically, but functionally. How do different brain regions, separated by centimeters of tissue, coordinate their activity to give rise to a thought, a memory, or an action? To answer this, scientists use techniques like Electroencephalography (EEG) and Magnetoencephalography (MEG), which listen to the brain's electrical or magnetic chatter from the outside.

But this is like trying to understand the intricate conversations happening inside a busy concert hall by placing microphones on the outer walls. A major problem immediately arises: the skull and brain tissue act as a conductive medium. This phenomenon, known as ​​volume conduction​​ or ​​field spread​​, means the electrical activity from a single, powerful source can spread out and be picked up by many sensors at once. It’s an electrical echo. If two sensors, $X$ and $Y$, are listening to two different, truly interacting brain sources, they might also both be listening to a third, loud, and completely unrelated source that just happens to be active.

This creates a maddening illusion. Because the field spread is, for all practical purposes, instantaneous, the common signal arrives at both sensors at the exact same time. When we analyze the correlation between the signals from sensors $X$ and $Y$, we find a strong, zero-time-lag relationship. It looks like a connection! But it’s a ghost—a spurious correlation created not by a true dialogue between the brain regions we are interested in, but by the electrical signal of an innocent bystander being smeared across our sensors. Standard measures of connectivity, like coherence, are easily fooled by this phantom, often reporting strong connections where none exist. How can we possibly hope to see the true, subtle conversations through this fog of spurious echoes?

Here is where the magic happens. The solution lies not in building a better sensor, but in looking at the problem through a different mathematical lens. As we've learned, the coherence between two signals is not just a single number; it's a complex number, with a real part and an imaginary part. And it turns out that nature has conspired in a most beautiful way.

The instantaneous mixing caused by volume conduction—this ghost in the machine—contributes only to the ​​real part​​ of the cross-spectrum between the two sensor signals. Why? Because an instantaneous relationship has no time delay, which means it has a phase lag of zero. A zero-phase contribution is, by definition, a purely real number.

On the other hand, genuine communication between two distinct brain regions takes time. The signal must travel along an axon and cross a synapse, introducing a small but crucial time delay, $\tau$. In the frequency domain, this time delay becomes a phase shift. And a non-zero phase shift is the lifeblood of the ​​imaginary part​​ of coherence.

Suddenly, the path forward is brilliantly clear. If the artifact lives exclusively in the real part, and the signal of true, lagged interaction lives in the imaginary part, we can simply... ignore the real part! By calculating the ​​imaginary part of coherence​​, we effectively put on a pair of mathematical glasses that makes the zero-lag ghosts of volume conduction completely invisible. We are left with a measure that is sensitive almost exclusively to interactions involving a genuine time delay.

This is an incredibly powerful idea. We accept a trade-off: we become blind to any true brain interactions that might, by chance, have a perfect zero-degree phase lag. But in exchange, we gain a tool that is wonderfully robust against the most pervasive and misleading artifact in non-invasive electrophysiology. It’s a prime example of how a deeper mathematical understanding provides a practical solution to a thorny experimental problem.

The insight that "time delay lives in the phase" has spawned a whole family of clever techniques. Imaginary coherence is just one member of this family. For example, neuroscientists also use a method called the ​​Phase-Slope Index (PSI)​​. The logic here is just as elegant. A true time delay $\tau$ between two signals creates a phase shift $\phi$ that depends linearly on frequency $f$, according to the simple relation $\phi(f) = 2\pi f \tau$. This means if we plot the phase of the cross-spectrum against frequency, the signature of a true delay is a straight line whose slope is proportional to $\tau$. An instantaneous artifact, with its constant phase of zero, has a slope of zero. So, by calculating the slope of the phase, PSI provides another robust way to detect time-delayed interactions while rejecting zero-lag contamination.

These phase-aware methods are versatile. They can be applied using the classic Fourier transform to get an overall picture of connectivity, or in conjunction with ​​wavelet transforms​​, which allow scientists to see how these lagged interactions evolve over time and across different frequency bands.

Furthermore, these tools are not mutually exclusive with other approaches. Scientists also try to tackle volume conduction by building sophisticated models of the head to mathematically "unmix" the sensor signals and estimate the activity back at the source locations in the brain. This is called ​​source reconstruction​​. But even these advanced methods are not perfect; some "leakage" of activity between estimated sources always remains. So, as a final layer of defense, a researcher can first reconstruct the signals in source space and then compute the imaginary part of coherence between them. It’s a beautiful one-two punch: first, try to remove the artifact physically, and then use a mathematical filter to catch any residuals that sneak through.

So far, our story has been about the brain. But the profound utility of separating a complex coherence into its real and imaginary parts is a truly universal principle. To see this, let's take a giant leap from the squishy, complex world of the brain to the pristine, quantum realm of a single atom.

Consider a simple two-level atom, with a ground state $|0\rangle$ and an excited state $|1\rangle$. We shine a laser on it. The laser field coaxes the atom into a delicate quantum superposition of the two states. The relationship between the ground and excited state components is described by a complex number, $\rho_{01}$, which physicists call the "coherence." Just like the coherence between two brain signals, this atomic coherence has a real and an imaginary part. Do they also tell different stories?

They absolutely do. By solving the fundamental equations of motion for this system, known as the optical Bloch equations, one finds something remarkable.

The ​​imaginary part​​ of the atomic coherence, $\text{Im}[\rho_{01}]$, is directly proportional to the rate at which the atom ​​absorbs​​ energy from the laser field. It represents the process of the atom "taking in" a photon and jumping to the excited state. It is the "in-phase" component of the atomic response that does work.

The ​​real part​​ of the atomic coherence, $\text{Re}[\rho_{01}]$, is related to a different phenomenon: ​​dispersion​​. It describes how the atom influences the phase of the laser light, effectively speeding it up or slowing it down as it passes. This is the "out-of-phase" component of the response.

Think about the parallel. In neuroscience, the imaginary part isolates a specific physical process: time-delayed communication. The real part is dominated by another: instantaneous field spread. In quantum optics, the imaginary part isolates absorption, while the real part isolates dispersion. In both cases, the decomposition of one complex number into its two orthogonal components allows us to cleanly separate two distinct, concurrently happening physical processes.

What began as a clever trick to clean up brain signals is revealed to be a manifestation of a deep principle woven into the fabric of physics. Whether we are studying the intricate dance of billions of neurons or the simple response of a single atom to light, nature uses the full richness of complex numbers to write its story. By learning to read both the real and the imaginary parts, we gain a much deeper and clearer view of the world around us.