To investigate the brain's topology, we have to understand a little about the issue of scale. A human brain scales over 9 orders of magnitude, from molecules that are only Angstroms big, to a cerebral cortex that spans around 15 cm. On this page we'll talk about the interaction between electromagnetic fields and neurons. Traditionally there are two somewhat opposite views of this relationship. Fields are continuous, and sometimes it's easier to treat a neural network mathematically as being approximately continuous. On the other hand, neurons are discrete, and so are their components down to the molecular level. So how is it that we get long-distance coherence in the fields that are generated by spiking neurons?

The first thing to understand, is that external electromagnetic fields DO affect neurons. There is a lot of conjecture in the literature about this effect being small or minimal - don't buy it, because it's not true. On this page we're going to quantify the relationship, and what we'll discover is, that neurons and electromagnetic fields have a mutual and computational relationship. Neurons generate fields that affect other neurons. Those fields affect everything from spike timing to the structure of the cytoskeleton. Most importantly for this discussion, they affect the behavior of neural networks near criticality. On this page we'll build a scaling model for the relationship between fields and neurons. If you're familiar with the theory of "volume conduction" in neuroscience, this discussion will ring a bell with you.

Let's begin on the continuum side. There is a physical limit called the speed of light, it's about 3 x 10¹⁰ cm/sec, in free space. In materials other than free space, the speed of an electromagnetic wave can slow down, for example electricity moves a lot faster in a wire than it does in a neuron, because the electrons in a wire move more quickly than the ions in solution. In a neuron, ions are the charge carriers, and different ions move at different speeds, depending on their charge, size, and other factors like the local environment. If an electromagnetic pulse starts at one end of the brain, by the time it gets to the other end it won't look like a pulse anymore, it'll be spread out with a funny-looking waveform. It reflects the nonlinear behavior of the material, which includes the neurons as well as the molecules.

If we just put an electrode on the scalp and measure the voltage and observe an EEG, we're picking up the collected activity of billions of neurons. These neurons have structure, they're neatly arranged into layers in many cases. The structure is a little weird though, the cortex is all curled up into sulci and gyri and it's not entirely clear what if any computational value this has. In many cases, geometrically opposed electric fields will cancel themselves out, but in other cases the geometry will cause neighboring fields to align, and in that case we can pick up a nice big signal on the surface. But what are we really picking up? If we put an electrode inside the brain we see something different - we see local extracellular field potentials, which exist at a somewhat different scale than the macroscopic EEG.

Let's anchor the discussion by referencing two research papers. The first is a long-awaited public recognition of the importance of fields, and the second is an observation about the coherence of field potentials in a geometric context. You'll notice the date on the first paper is 2025, just a year ago. That's a long time since Hodgkin and Huxley showed us how an axon works, and an even longer time since Luigi Galvani discovered "animal electricity". Research into Trans-Cranial Magnetic Stimulation (TMS) and Deep Brain Stimulation (DBS) proves beyond any shadow of a doubt that neurons respond to external electromagnetic fields. Recently, Earl Miller's group highlighted how the influence of fields extends all the way down to the molecular level. This important paper observes a depolarization of 0.5 mV in a neuron, when another distant (and unconnected) neuron fires. It gets even more dramatic: synchronization can be elicited in hippocampal slices by certain kinds of stimulation. When the slices are cut, the synchronization persists - even when the slices are completely separated by an air gap.

What does this mean? What does this tell us? Watch this video. The pendulums synchronize because little bits of energy get transferred between them, along the attachment points. Similarly, fields transfer little bits of energy between neurons, encouraging them to synchronize. In some cases (like the air-gap hippocampus), the fields are sufficiently strong to maintain the coupling even when the neurons are disconnected. In the cerebellum, the fields generated by Purkinje cells are enormous, their axon initial segments are large, and ephaptic communication between neighboring Purkinje cells has been repeatedly demonstrated. So then, if fields can affect neurons, the converse must also be true, which is neurons can affect fields. Therefore there is a mutual relationship between neurons and fields. What exactly is the nature of this relationship?

For most physicists, the concept of a neural field is a whole-brain process. For example in a Hopfield network, the Hamiltonian carries with it the assumption that it completely describes the network. The energy surface is a unit, it's defined on the whole network. But there are also convolution networks, with very small filters, which in some cases can be equated with topographic connection maps. And, both the theory of receptive fields and the theory of population dynamics tell us that the small filters will combine into bigger filters, and the result is contrast enhancement and gain control and a host of useful behaviors. An important piece of understanding comes to us from the theory of dynamic neural fields, which is based on population dynamics. DNF treats a neural network as a continuum, it's very similar to the original Wilson-Cowan concept. In a continuum, you have a surface - in a discrete setting, the surface becomes a mesh. A mesh is a discrete approximation to a continuum, and if you believe in connectomics, the continuum is itself a continuous approximation of a discrete structure. But this view is completely dependent on the concept of a unitary synapse, and the existence of fields diminishes the connectionist view. Why do I say that? Let's look at the second research paper.

This paper uses electrode arrays to measure the coherence of signals in the cerebral cortex at multiple levels of resolution. The authors looked specifically at the primary visual cortex, which is a curved structure at the back of the occipital lobe. They discovered that the coherence between spike trains followed the geometry of the sulci and gyri, and not the connection map of the neural network. Once again we find a dissociation between the expected relationship between neurons and fields, and the actual relationship. Some in the machine learning community would have us believe that neurons only care about synapses and weights, and clearly this is not the case. So what kinds of interactions follow structural geometry rather than connectionist geometry? Well, there are several. Astrocytes, for one, are excellent candidates. As are fields, which are only partially aware of the connectome. Connectionist theory tells us that for any input, there will be "hot spots" where amplitude drives an earlier and stronger neural signal, which is then moved laterally through inhibitory interneurons. And population dynamics tells us that all this will occur on one of several computational platforms depending on the current state of the multi-stable network. It could be damped and linear, in which case a strong input will create a strong output followed by a few oscillations that eventually die down. Or, it could be in an oscillatory regime where the input can affect both the phase of the population oscillation and the timing of individual neuronal spikes. Or it could be in a critical regime at the edge of two states, and the field could be sufficient to drive it one way or the other. Each of these modes could involve bursting.

So what determines the boundaries of a signal in the brain? If it's moving laterally at the same time it's moving forward, it's kind of hard to tell where it really is. If there are two equally strong signals, will there be two processing regions or will they combine into one gigantic representation of the input? These are some of the questions that DNF tries to address, as it's essentially the population dynamics of spiking neurons. In the hippocampus, there are "sharp wave ripples" that engage about 100,000 neurons at a time, and curiously enough, the domain of an astrocyte is about 100,000 neurons. In the cerebral cortex there are processing mini-columns about 100 microns wide containing about 100 neurons and an equal number of astrocytes, and there are also columns with a 300-400 micron extent and hypercolumns that organize over 1-2 mm. On many levels, it looks like there is a hierarchy of scale - almost like a mesh within a mesh, where the inner meshes are being progressively refined. Which is an absolutely perfect architecture for multi-window causal analysis, but what else does it give us? It would seem that the algebra is being "confused" by the oscillations, by the synchronization, and by the fields. At least, it somehow doesn't look like the classic machine learning model where A_t+1 = Σ Wij * A_t.

In the continuum view, the fields generated inside the brain encounter a boundary in the bony tissue around the skull, including the meningeal layers that protect the cortical surface. For this reason, the boundary is often modeled as a sphere, the brain is placed inside the sphere, and fields that encounter the boundary either bounce off (are reflected) or are absorbed (or something in between, for example the semi-absorptive wall is well known in acoustic modeling). A spherical boundary is friendly because it lends itself to both analytic and numerical solutions, and in many cases if the shape of this boundary were to change the equations would break down (since they're based on assumptions of isotropy and homogeneity).

Working the other way, we can begin with the discrete view, which consists of a bunch of field generators on a geometric lattice (or a mesh, in the unstructured case). Let's consider this view in some detail. The discrete view has the implicit assumption of "fixed" geometry, which is almost never the case in real brains. But let's note that and move on. The question is, at what levels can the nodes communicate? They can certainly communicate with nearest neighbors, and anything they happen to be connected to, but what about things that are far away? Let's do a set of thought exercises. According to Earl Miller's team a small axon not too far away can generate a depolarization of 0.5 mV in another axon. So, how many "small axons not too far away" would it take to drive this neuron to threshold? Let's say the resting potential is -70 mV and threshold is -20 mV, so we need 50 mV worth of stimulation, which it turns out, is only 100 neurons. And let's put that into context. Any two pyramidal cells in the cerebral cortex can have 1000 or more synapses between them, and there are approximately 100 neurons in a cortical mini-column. What are we really talking about when we say "synchronization"? Well, we're mostly talking about coincident spike trains in neurons that are several hundred microns apart. So we're not talking about the neurons in a column, instead we're talking about synchronization between columns, and more specifically between the layer 5 pyramids we're measuring. And what we already know about columns, is they oscillate. Just about 100% of the time. Sometimes they can oscillate in synchrony, and sometimes they look more like a bunch of Chua devices ("aperiodic" oscillators).

So even though it has a fixed geometry, our lattice is operating on multiple spatial and temporal scales. What does a neural oscillation actually look like, in terms of an electric field? Well, the speed of light in seawater (which is only slightly more ionic than CSF) is around 3/4 of what it is in free space, which puts the wavelength of an alpha frequency around 25 million meters. So in a brain, the field is flat, that is to say, in the absence of any intervening structure, it has a radial symmetry. To say it a different way, 100% of the distortion is due to the intervening geometry. Which is a wonderful observation, because it makes the entire structure computationally accessible. One has to be very careful with the simulators, because traditional simulators like Neuron and Brian2 handle this completely bass-ackwards, they calculate the extracellular potential from the local currents. Which may be a good enough approximation to mimic some behavior but it's not real. To be realistic, one has to take the fields into account, because the fields provide a distinct pathway for communication between neurons. Fields have a completely different scale than synapses do. Synapses are limited by the transmission time along axons, but fields are not. And in all fairness, we should also mention there are slower processes too, like astrocyte mediated calcium events that occur on a time scale of seconds rather than milliseconds. So on our lattice, we have not only the geometric positioning, but we have multiple forms of topological connectivity, each of which has geometric data associated with it. (Earring, anyone?)

Now we're starting to see the real nature of this so-called "discrete" network. It's not really a discrete network at all, what happens here is there is a discrete embedding into a continuous domain. And that domain, is very odd looking, it has some very strange topological properties that are handed to it by the underlying structure. It's not a point topology, it's not a "topographic point to point mapping". It's a loop topology, which is something entirely different. The loops occur at multiple scales, and they have different shapes and different representations, and they all interact with each other. This is a much harder math problem than just multiplying a couple of matrices! In fact, this kind of math approaches what they call "intractable", because even if you could find an analytic solution (which you can't), you still couldn't compute it. The good news is, you can numerically approximate it, with multi-physics. What is "multi"-physics? Well, it's exactly what it sounds like, it's structural mechanics, fluid dynamics, and charge all rolled into one. It isn't exactly a "complete" description of a small volume of brain, because we're leaving out organelles and molecular motion and a lot of other things. What we're really interested in is the relationship between the fields and the ions. And that relationship is accessible, through nonlinear thermodynamics and especially via stochastic relationships between physical variables.

We could do molecular dynamics on this scenario, that's within the realm of possibility. There are some excellent open source molecular dynamics programs like LAMMPS that will leverage all the available computing resources on complex domains. But then what would we see? A bunch of molecules floating around in a membrane? Not so sure that helps us... what we want is the field effect on ionic currents, which involve a very special set of trans-membrane ion channels, which is what we're modeling in the traditional mesh. To expand the traditional mesh to accommodate field effects, we only have to include the electric and magnetic fields in the equations. How do we do that? Let's say we have a Hodgkin and Huxley equation, how do we incorporate the effect of an extracellular field? We can't just add an extra current term, because we don't yet know how the fields affect the currents. We can make assumptions about that, but until we're able to engineer the behavior we can't say we understand it.

Conversely, since we don't know much about these fields, the issue of stimulus delivery becomes important. In a normal experiment we would orchestrate the stimuli in a controllable manner, but here we can't really do that. We don't want widespread stimulation, no DBS or TMS, we need a localized source. At minimum we need a "pulse" we can measure, something that approximates a Dirac delta function. This way, we can observe the impulse and steady state responses at the same time. And we also need a way of quantifying the pulse, in other words does the effect drop off as the square of distance, inverse of the square, or what? We'd like to fix the pulse at a known time and position, so we can quantify the effects using the method of variations. The only way we can realistically do this with today's software is using a finite element or finite volume method, because the fields affect every intra- and extra-cellular compartment and every membrane. Ideally we'd like a resolution that allows us to look "inside" a synapse, but not so small that we can't do statistics on it. Something in the 25 nm range sounds reasonable, because it's bigger than a molecule and smaller than a synapse. Within that space we can count enough ions to make statistical approximations.

Thinking about the scale of that, a 25 nm mesh on a whole neuron consists of well over a million vertices, and at least twice that many faces. If we put a few of these neurons into a network, we're looking at billions of computational nodes. Most software can't handle that, certainly not the software we usually find on the desktop. We're no longer dealing with toy cable models or ball-and-stick neurons, we're dealing with real fields on real geometry. The simulation is on the same scale as molecular dynamics, we'll need to generate billions of random numbers to account for the stochastic behavior of ion channels. Is there an easier less demanding way to model this? Not really. The existence of electromagnetic fields in our model precludes any pretensions to simplicity. We need to recognize this reality up front. Realistic neural modeling is not matrix multiplication. The simulations take hours, sometimes days. The finite elements are finicky, the solvers can go north even with a perfect mesh. What we specifically want to avoid, is making destructive assumptions in favor of computational speed. This issue has plagued the simulation community from day one, and the good news is that in today's world, we have near-supercomputer capability available to us in the cloud, and the desktop is much better than it was 20 years ago. The time is right to revisit this approach and learn something from it.

From single ions we've already jumped to molecules and membranes, and the next step is dendritic arbors and astrocytes, and after that we finally get to the connectome. That's four levels of scale already, and we've already talked about mini-columns, columns, and hyper-columns, that's three more levels, and then we can talk about long range connections and the corpus callosum - so this is a pretty rich simulation already, but so far it's scaling beautifully. The relationships are hierarchical at each level, except for one thing: fields cut across levels to affect all elements simultaneously. In a way they're like the external field in an MRI that makes the atoms precess, we get information from that and turn it into meaningful images. Now, let's get back to the question of what determines functional boundaries, because we have some new information.

The loop topology is a convenient way of embedding networks into other networks. Done procedurally, this results in a recursive algorithm and the resulting geometry is self-similar (it's "fractal"). The interesting question is how this all relates to the traditional view of neural networks as synaptic matrix calculators. Let's do another thought experiment. Let's say we have a mesh, a regular 3-d lattice. In a machine learning setup, we have the feed-forward path and the backward path, and discrete time steps, and in each time step we're going to sweep forward (generating errors) and backward (correcting them). In such a setup, the shape of the memory is determined by the shape of the connections. In a Hopfield network where everything is connected to everything else, the memory ends up being distributed over the entire array. Whereas in a plastic convolutional network, the connectivity is such that we're restricted to local covariances, and in such a network the local dynamics become influential, with or without global participation. In a traditional (connectionist) neural network simulator, information is carried between neurons "only" by synapses. There are some variations, like gap junctions and so on, but that's the basic idea behind connectionism. Therefore with binary neurons that can be either firing or not firing, there are 2ⁿ network states. How many network states are there, in the equivalent continuum? Obviously, there are infinitely many. In several kinds of physics, the relationship between "many" and "infinitely many" is described in terms of functionals, which is exactly what we're trying to solve for with a finite element model. Is there such a thing as a "regional Hamiltonian"? What do you think? And if there is, is there continuity between neighboring regions or do they operate independently?

Here's one of the important questions: can we access the entire global memory store from just one computational subunit? Can we somehow make a neuron "read out" a portion of memory? The same question exists at the network level, can we encode a "subspace" of the global store into a sharp wave ripple? Using phase encoding or any other method? Apparently, the answer is yes, provided that certain conditions are met. The network must be in an oscillatory regime, it won't work with static attractors. Which immediately suggests that phase coding could be important, and in fact it's the perfect method for such a mechanism. Phase coding is powerful, it can be used in several important ways, for multiplexing, for feature extraction, for data compression. And we just learned on this page that synapses are only one of the many ways neurons can communicate. Is there phase coding at the level of the field? That's an interesting question because neurons operate on a scale of milli-seconds, they generally can't reach the nanoseconds involved in a field. However the field affects the timing (phase) of spike trains, and spike trains in a population most certainly can achieve nanosecond resolution. What is the view from the field perspective? At the boundary of a model sphere that encloses our volume, there is low-level activity and a gentle sea of waves, until something starts bursting. Then there are rapid perturbations in the field. When lots of rapid perturbations occur in phase, we get visible traveling waves, which is exactly what we see in the EEG. The pattern of bursting can be considered relative to the field, that is to say, relative to all the activities of nearby neurons. In this way phase encoding becomes "regional", and the boundaries of regions can move just like they do in dynamic neural fields. This mechanism provides a robust computational link between the field configuration and the geometry of ion densities near a neural membrane.

Achieving scale on a mesh is pretty easy (if the mesh tools work), but the idea of scaling over 9 orders of magnitude is daunting computationally. First it likely requires 128-bit floating point representation to prevent underflows when multiplying small numbers, and in today's computing environment there is no support for 128-bit math at the workstation level. This capability mostly exists at the supercomputer level (which is why most of the published simulations happen on supercomputers). However there are ways of making it work, with ordinary 64-bit math. One can certainly emulate the 128-bit math in software, then everything slows down, and many people are too impatient to wait for the results, so they end up sacrificing accuracy for computational convenience, which kind of defeats the purpose of simulation. We get some very small numbers in simulations, for example conductivities are often delivered in terms of picoSiemens, and to keep the solvers happy we can't allow any divide-by-zero scenarios, or even divide-by-near-zero. The 128-bit math give us 38 decimal digits, which is good enough for our multi-physical purposes.

In the rest of this section we'll look at various kinds of simulations. If you have an experiment in mind and you're looking for a set of tools to connect your workflow, you have essentially three choices. You can use an existing macroscopic system like Allen or OBI, in which case you have to conform to their workflows. You can collect working tools yourself, in which case you can expect to spend some time surfing Google and GitHub and discovering which tools actually work. And you can start from scratch. Which sometimes... well let's just say, it seems like an attractive option. Unfortunately in today's world, there's plenty of AI hot-shots who think they can use Claude to convert CGAL, and usually they'll make the one piece they need work, and neglect to test the rest. And then post the result to GitHub and use it to get a job, and that's the last we ever hear from them, they disappear into the black hole of the corporate world. The result is there's a lot of untested garbage in the public domain, and if you're just starting out I'm very sorry but that's reality. However I can help. Maybe I can help save you some time.

Since I'm engaged in this work on a daily basis, I get to see what works and what doesn't, so I've created a web site called "neural-modeling.org" that catalogues toolsets on the basis of their functionality. If something installs okay and does what it says, it gets a thumbs up. If it won't install, isn't being maintained, or doesn't do what it says, it gets a thumbs down. Hopefully it will save you some time in your search for the tools you need. Neuroscience is a vast field, the need for tool sets never ends, and constantly changes. The best thing software can do is put some stakes in the ground, so people who need the same functionality don't have to keep reinventing the wheel. From a scientific standpoint there is a set of standards called FAIR that software support needs to adhere to, and these standards require active maintenance, so someone who needs to reproduce the results twenty years later can still find the datasets and the toolsets. In real life this has mostly to do with money, when grants terminate there's no more money left for things like certificates and IT administrators, and the result is scientists have to become programmers, they have to build the tools they need themselves. That's certainly true in the case of finite element modeling for neural fields, it's an esoteric subject and the tool sets that handle nuclear fusion simulations are not the same as the ones neuroscientists need. You can see one of the tools I'm working on at annie-interface.org, it's a workflow that brings finite elements closer to the desktop.