The magnetic field generated by an infinitesimal line current element in a sphere

Humans can capture with their eyes a tiny fraction of the total information carried by the electromagnetic waves around us. Take a minute and think of how many things we know that exist because we experience their results/effects, but we cannot really see them. For example, floating all around us there are radio waves, Wi-Fi signals, sound waves, communication signals, radiation etc. All of this invisible information is a composition of oscillating electric and magnetic fields propagating through space as waves described by a set of four equations discovered by James Clerk Maxwell and published in 1865. Speaking about electromagnetic waves, let us take a closer look on magnetic fields.

A magnetic field is a vector field, i.e. a property of a region of space that has magnitude and direction. Maxwell’s equation can be used to compute through mathematical operations the magnetic force experienced by electric charges or electric currents moving through the space where the magnetic field is known. In a less mathematical way, the magnetic field is an emerging property of the region of space around a magnetic material or a moving electric charge which influences charged bodies and other magnetic materials as what we call a magnetic force. Magnetic fields surround magnetized materials and are also created by electric fields varying in time. Since the strength and direction of the magnetic field changes depending on the location, the magnetic field is described using a 3D vector at each point in space, called a vector field. The Earth itself produces its own magnetic field and is in fact one of the strongest magnetic fields that we can measure, with a magnitude ranging from 25 to 65 μΤ at its surface. Magnetized materials, wires carrying electrical current and electric charges moving in space through time also generate magnetic field around them. One other source of magnetic field not as obvious as other sources of magnetic fields is the human brain. Yes! The brain, one of the most complex organs in the human body.

The human brain is made up of billions of neuron cells. These neurons communicate with each other through synapses and their main function is to relay and process the input-sensory information allowing us to move, think, feel, store memories, and in general be able to coordinate and control our actions. When information is being processed by brain neurons, small currents flow in the neuronal fibers thus, producing magnetic fields. Although, these magnetic fields are much weaker than the Earth’s magnetic fields they can still be detected noninvasively thanks to a relatively novel functional neuroimaging technique called magnetoencephalography (MEG). The development of the MEG neuroimaging technique has been a breakthrough in the field of neuroscience due to the new information that it offers about the functionality of the human brain. Compared to other neuroimaging techniques, MEG has high spatiotemporal resolution and the measured magnetic signals are not affected by the high electrical resistivity of the skull. To briefly describe MEG, an array of MEG magnetometers (~ 300 sensors) is arranged in a helmet shaped vacuum flask located around 2cm above the surface of the head. An MEG experiment is the recording of the magnetic fields (by the MEG sensors) generated by the human’s brain-neuron cells while the human (the subject) receives some type of sensory stimulation, or it can be resting state recording where there is not any stimulation. The output of an MEG recording is one time-series signal for each one of the MEG sensors. These data can either be processed analyzed as they are (in sensor space) or they can be used to estimate the activity inside the brain (source space) by solving the MEG forward and inverse problem. The MEG forward and inverse problem will be introduced in more detail and explained in one of my next blogs. For now, we talk more about what is measured by the MEG sensors and try to understand it through a very simple model and simulation.

What is actually measured by the MEG sensors above the surface of the head is a linear combination of the magnetic fields generated by the flows of current in different bunches of neurons (neurons having similar orientation and close to each other) located in different brain regions. The detectable magnetic field is not generated by a single neuron cell itself, this is too small to measure by any available instrument today. If many neurons are randomly oriented and fire asynchronously, no matter how many they are, they will not generate appreciable magnetic field. The effect of any individual neuron will be two small and on average activity generated by randomly oriented sources will cancel each other out. To produce a measurable magnetic field the collective activity is needed from many neuron cells close to each other, having the same orientation and firing synchronously. The existence of more than one of such synchronized neuronal populations results in more than one magnetic field sources. For this reason, it is not easy to disentangle and localize the different sources from the topography of the measured MEG signals. This is what the inverse problem attempts to do. The good news is that there are quite a few methods and algorithms that can do that with good spatiotemporal resolution. It should be no surprise to hear that to solve the inverse problem one requires as input the solution of the MEG forward problem. This is finding what magnetic field is produced when a known source of magnetic field placed in some location and for each one of its possible orientations. The inverse problem then becomes an optimization problem: finding the best combination of such elemental sources that will produced a given set of MEG measurements.

When dealing with a complex data as the ones generated by the brain, is not enough to use techniques as black boxes. It is wise to have a good understanding of the law of physics and the basics principles from electrophysiology and anatomy that the generation of these signals must obey. As a first stage, it is very important and helpful to have an idea of how the magnetic field should look like based on the location and orientation of its source in the brain. In this way one can evaluate the results of the source localization techniques and be able to say if they make sense, be able to identify the source of the problem and even tune the parameters of models and the algorithms accordingly.

In this blog, we use a simple model to simulate and visualize the magnetic field generated by a single source in the brain. The brain is modelled as a homogenous and isotropic conducting medium with the shape of a sphere while the hypothetical neural source is an infinitesimal line element of current located inside the sphere’s volume with a specific orientation and direction. The magnetic field is measured on a 2D regular grid located above the top surface of the sphere with in-between space of 1/8 the radius of the sphere. To understand the influence of the sphere’s bounds and symmetry we also estimate and compare the magnetic field that is generated by the same source in a homogeneous free space (see figure below).

 

For the calculation of the magnetic field outside the sphere we use the equation derived from the Biot-Savart Law for an infinitesimal line element of current in a spherically symmetric conductor which is relatively simple due to the symmetry of the sphere. Similarly, the magnetic field generated by the source in the free space is estimated by the corresponding equation derived again from the Biot-Savart Law. The only inputs to these equations are the radius and center of the sphere, the location and intensity of the current source and the coordinates of the point in space we want to estimate the magnetic vector. By estimating the magnetic vector in all the points of the plane we finally have the magnetic field on the 2D plane above the sphere and in free space. The video below shows the magnetic field measured on the 2D plane in the case of the source being in the sphere and in the free space, for different locations while keeping its intensity and orientation fixed. In order to see the effect of increasing the distance of the source from the measurement plane, we move the source in the Z-axis from the highest to lowest point of the sphere, passing through its center. The orientation of the line element was chosen such that it produces a symmetric dipolar pattern around the X-axis and Y-axis.

In the video shown below, the two plots in the left column show the z-component of the magnetic field at each of the points on the measurement plane. The top plot corresponds to the magnetic field when the source is in the sphere and the bottom plot in the case where the source is in free space. The percentages in each of the plot show the ratio between the maximum z-component of the magnetic vector at the current location of the source and the maximum value from all the locations (which is the top position).  The source location in the sphere is shown in the top-right diagram, the source is moving on the Z-axis from the highest to lowest point of the sphere passing through its center. Note that in this video, the orientation of the current line element should point exactly vertical and outwards the screen, however it is plotted in this way just for visualization purposes. Finally, to compare the magnetic field in the case of the source being in the sphere and in the free space, we plot the maximum z-component of the magnetic field for each location of the source for the two cases (bottom-right plot).

 

From just a simple model, some useful conclusions can be made. Firstly, it is clear that the magnetic field in the case of the sphere-model changes polarity when passing the center of the sphere while the polarity remains the same when the source is in free space. Secondly, from the bottom-right plot we can see that the intensity of the magnetic field drops rapidly when the source approaches the center of the sphere and is exactly zero at the center while it has a constant decrease in the case of the free space. This effect at the center of the sphere is caused by the fully symmetric shape of the sphere conductor model. In one of the next blogs, we will show what happens when this symmetry breaks either by passing just near the center or by using a non-symmetric conductor model. After passing the center of the sphere and moving downwards, the intensity of the magnetic field increases again approaching asymptotically the intensity of the source in free space. Another important observation is that the dipolar pattern of the magnetic field is more focal in the case of the sphere model. In addition to that, the distance between the centers of the positive and negative values increases faster and more when the current element is in free space. These two last conclusions together are telling us that the sphere model somehow works as lens, focusing and concentrating the generated magnetic0 field from the other side of the sphere center.

Every human’s head shape can be indeed approximated as a sphere, some better than others. Some elements of the general observations we made based in modelling the magnetic fields generated by an infinitesimal line element in a conducting medium inside a container with perfect spherical inner boundary of low conductivity, will be present in real data collected from human subjects. However, if we take a more detail description of the head’s shape, we can see that it is not completely symmetric. This small deviation of the real head shape from the completely symmetric sphere shape can make a difference, especially for the detectability of generators placed at the centre of the head. For a skull which is perfectly spherical, no matter how strong a generator is, it will produce no external magnetic field outside the sphere, if it placed at the exact centre of the sphere. This is not any more true if the exact symmetry is broken, even by a small amount.

But how are all these related with the i-CONN network and connectivity science? In this blog, we use a very simple model for simulating the magnetic fields generated from the activity of the neurons in the brain. Even with this simple model, we were able to understand the basics of the magnetic field properties, visualize the effect of the head’s shape symmetry on the ability to measure the magnetic field and understand the importance of breaking this symmetry using a more realistic model of the head’s surface. Estimating the connectivity between different regions in the brain and being able to derive conclusions about the flow of information, demands having the brain’s activity distribution with high spatiotemporal resolution and accuracy. The simple simulation presented here, shows an important fact that affects the accuracy and resolution of source localization using MEG data: the availability of a good description of the conductivity model of the head and especially (for MEG) the exact shape of the inner surface of the skull.

The main message of this blog is that if we really want to improve something its important that we take a step back to the basics and trying to understand the underlying physics and the modelling of processes that determine the behavior of the system we want to understand.  The ability to go to first principles is often the beginning of the correct road to the solution.

Thank you, I hope you find this blog insightful and interesting, stay tuned for the next blog for even more interesting physics, simulations, neuroimaging and connectivity science….!!!