Coupled Antennas: Literature on Small Separation Distances

[JPBenowitz]

Can anyone provide some literature on antennas spaced apart such that the spacing distance is much less than the wavelength? I am specifically interested in the mutually coupling behaviour as a function of small separation distances.

 

[Baluncore]

They would not be considered as individual antennas if they were that close.
They would be analysed as a single antenna.

This is a numerical job for the NEC style of antenna simulators.
http://en.wikipedia.org/wiki/Numerical_Electromagnetics_Code

You will need to better identify their structure before an analysis of the specific case can be referenced.

 

[davenn]

Can anyone provide some literature on antennas spaced apart such that the spacing distance is much less than the wavelength? I am specifically interested in the mutually coupling behaviour as a function of small separation distances.

you really need to give a bit more info on what you have in mind
1) are the 2 antennas on the same freq?
2) do you want them as a phased array ?
3) what sort of antennas ? ... dipoles, yagis?

for phased yagis, to get the best gain and radiation pattern, the optimum distance apart for a phased array is one wavelength
any closer than that and the interaction between the 2 arrays destroys radiation patterns which will result in loss of gain.


cheers
Dave

 

[JPBenowitz]

you really need to give a bit more info on what you have in mind
1) are the 2 antennas on the same freq?
2) do you want them as a phased array ?
3) what sort of antennas ? ... dipoles, yagis?

for phased yagis, to get the best gain and radiation pattern, the optimum distance apart for a phased array is one wavelength
any closer than that and the interaction between the 2 arrays destroys radiation patterns which will result in loss of gain.


cheers
Dave

I am modeling neuronal axons as Collinear Coaxial Cables that are separated on the magnitude of nanometers to micrometers where the wavelength is on the magnitude of millimeters.

 

[Baluncore]

I believe you might treat parallel axons as close directionally coupled coaxial cables.
If I remember correctly, like a coaxial cable has an insulated external conductive screen, an axon has a myelin sheath.

If the chemical charge balance remains within that myelin sheath structure, then there should be little electrostatic imbalance. If the longitudinal charge transfer is balanced then there should be little external magnetic field.

That would suggest that axons should not cross-couple. They are natural coaxial cables.
So what have I missed? What false assumptions have I made?

 

[JPBenowitz]

I believe you might treat parallel axons as close directionally coupled coaxial cables.
If I remember correctly, like a coaxial cable has an insulated external conductive screen, an axon has a myelin sheath.

If the chemical charge balance remains within that myelin sheath structure, then there should be little electrostatic imbalance. If the longitudinal charge transfer is balanced then there should be little external magnetic field.

That would suggest that axons should not cross-couple. They are natural coaxial cables.
So what have I missed? What false assumptions have I made?

There are myelin gaps every 2mm along the axon with a high density of ion channels. When the channels are active there is both an influx and efflux of ions across the membrane. So we have ions from the electric double layer (the effective outer conductor) diffusing through the unmyelinated membrane into the cytoplasm (the effective inner conductor) and ions diffusing from the cytoplasm through the unmyelinated membrane to the electric double layer. I've have been modeling this as a Collinear Coaxial Cable where at each myelin gap the inner conductor is connected to the outer conductor.

 

[Baluncore]

A large magnetic impulses can cause significant currents in an electrolyte, which may trigger an electrical response. But I don't see how an axon can generate sufficient magnetic field to influence a neighbour. I think an electrical model that ignores magnetic fields may be quite realistic.

The myelin sheath can be seen as the common black insulation on a coaxial cable. I suspect the diffusion channels at the nodes are used to balance the electrolyte levels within the axon.

So consider a model where there is a matrix of resistors at each node. That matrix represents the ion flows across the node, between layers and with the external ion reservoir. That external ion reservoir can probably be modeled as an electrical ground.

If the gaps in the myelin sheaths of parallel axons line up with each other, then there may be some slight coupling or modulation between nodes by ground potential differences, due to ground currents. But I expect the “internal” signals are sufficiently “digital” to not be significantly effected.

If we think of a node as a resistive “T” or “Pi” attenuator, then there must be sufficient gain in the chemical transmission line, (axon), to regenerate the pulse after attenuation through a node. I guess diffusion at the node must also provide the chemical energy to power that gain.

What would the resistive matrix modelling a node look like? What ions are flowing?
The more I think about the model, the less I see it as coupled antennas, and the more I see it as a network of currents.

 

[JPBenowitz]

A large magnetic impulses can cause significant currents in an electrolyte, which may trigger an electrical response. But I don't see how an axon can generate sufficient magnetic field to influence a neighbour. I think an electrical model that ignores magnetic fields may be quite realistic.

The myelin sheath can be seen as the common black insulation on a coaxial cable. I suspect the diffusion channels at the nodes are used to balance the electrolyte levels within the axon.

So consider a model where there is a matrix of resistors at each node. That matrix represents the ion flows across the node, between layers and with the external ion reservoir. That external ion reservoir can probably be modeled as an electrical ground.

If the gaps in the myelin sheaths of parallel axons line up with each other, then there may be some slight coupling or modulation between nodes by ground potential differences, due to ground currents. But I expect the “internal” signals are sufficiently “digital” to not be significantly effected.

If we think of a node as a resistive “T” or “Pi” attenuator, then there must be sufficient gain in the chemical transmission line, (axon), to regenerate the pulse after attenuation through a node. I guess diffusion at the node must also provide the chemical energy to power that gain.

What would the resistive matrix modelling a node look like? What ions are flowing?
The more I think about the model, the less I see it as coupled antennas, and the more I see it as a network of currents.

That's the contemporary model utilizing Linear Cable Theory and its assumptions. The only problem is that is has been observed that neurons can synchronize their action potentials when they are close (See Ephaptic Coupling of the drosophila fruit fly). Furthermore, it has been most recently observed that separate parts of the brain can oscillate at the same frequency and is independent of electrochemical signals, in other words the endogenous electric field in the extracellular medium is assisting in information processing.

To start things off I do not model the axon as an equivalent circuit; this is an idealization that I believe has missed a fundamental property of the neuron. Instead I begin with the electrodiffusion equation and the poisson equation in cylindrical coordinates. This is a system of nonlinear 2nd order spatiotemporal partial differential equations. Conceptually the problem is easy. There is initially a Boltzmann distribution of ions on the surface of the myelin sheath (dielectric) and a Boltzmann distribution of ions on the inner membrane surface. Keep in mind the ionic species and surface charges are not the same. When we run the system in time there is active diffusion across the membrane perturbing the distribution of ionic species on both sides of the axon producing a time changing potential. The magnetic field has been measured and it is negligible but my hypothesis is that the far-field plays some role in neuronal oscillations.

Since there is a high density of ion channels and pumps at the myelin gaps and a sufficiently large potential difference across the membrane, when these channels open the ions will accelerate and thus radiate.

 

[Baluncore]

Sorry for the delay. It has taken some time to sort out what to model here. I think it best to consider the EM model from the viewpoint of antennas and the method of moments used to model coupled EM structures.

I note that the dielectric constant of water is about 80. For water trapped in tissue a constant of about 40 is more applicable. This gives us a typical velocity factor for the propagation of EM energy through tissue of 1/Sqrt(40) = 0.158, that is 15.8% of the speed of light in space. The phase velocity in tissue will be about 47.5 Mm/s in tissue.

The myelin sheath of the axon has a lipid sleeve that is an electrical current insulator. Lipids have a dielectric constant closer to two. The velocity factor will therefore be about 1/Sqrt(2) = 70% that of light. Relative to the surrounding wet tissue, the sheath will therefore make a low capacitance electrostatic screen and so should better isolate an axon from surrounding structures. This velocity relationship is also interesting because it suggests that any EM leakage will not be guided by the sheath, but disbursed widely and lost into the surrounding tissues.

The significant cyclic movement, or reversal, of chemical signals in axons have frequencies below about 10 kHz. We can calculate the wavelength in tissue for EM radiation from axons as 47.5Mm / 10k = 4.75 km. That indicates that coupling inside a body or between nearby bodies will always be in the near field. The near field boundary is defined as 60 wavelengths, which here is 60 * 4.75 km = 285 km.

So what does this tell us about coupling between axons in the brain? It tells us that they are all very short dipoles in the very near field. The radiation from any 2mm long section of an axon will be a simple dipole pattern. It will also be incredibly small. For that reason they can definitely not direct energy like a beam antenna.

Now we must consider the electrostatic pattern of an electrochemical impulse in an axon. The charge movement is not along the axon, but radially symmetrical about it's centre. For that reason, there can really be no radiated signal since the net direction of charge movement cancels to zero.

This radial cancellation is also true at the gaps in the myelin sheath.

Based on this analysis, I believe we can discount any EM connection between areas of the brain.
I believe any coupling between areas of the brain must be along axons.

 

[JPBenowitz]

Sorry for the delay. It has taken some time to sort out what to model here. I think it best to consider the EM model from the viewpoint of antennas and the method of moments used to model coupled EM structures.

I note that the dielectric constant of water is about 80. For water trapped in tissue a constant of about 40 is more applicable. This gives us a typical velocity factor for the propagation of EM energy through tissue of 1/Sqrt(40) = 0.158, that is 15.8% of the speed of light in space. The phase velocity in tissue will be about 47.5 Mm/s in tissue.

The myelin sheath of the axon has a lipid sleeve that is an electrical current insulator. Lipids have a dielectric constant closer to two. The velocity factor will therefore be about 1/Sqrt(2) = 70% that of light. Relative to the surrounding wet tissue, the sheath will therefore make a low capacitance electrostatic screen and so should better isolate an axon from surrounding structures. This velocity relationship is also interesting because it suggests that any EM leakage will not be guided by the sheath, but disbursed widely and lost into the surrounding tissues.

The significant cyclic movement, or reversal, of chemical signals in axons have frequencies below about 10 kHz. We can calculate the wavelength in tissue for EM radiation from axons as 47.5Mm / 10k = 4.75 km. That indicates that coupling inside a body or between nearby bodies will always be in the near field. The near field boundary is defined as 60 wavelengths, which here is 60 * 4.75 km = 285 km.

So what does this tell us about coupling between axons in the brain? It tells us that they are all very short dipoles in the very near field. The radiation from any 2mm long section of an axon will be a simple dipole pattern. It will also be incredibly small. For that reason they can definitely not direct energy like a beam antenna.

Now we must consider the electrostatic pattern of an electrochemical impulse in an axon. The charge movement is not along the axon, but radially symmetrical about it's centre. For that reason, there can really be no radiated signal since the net direction of charge movement cancels to zero.

This radial cancellation is also true at the gaps in the myelin sheath.

Based on this analysis, I believe we can discount any EM connection between areas of the brain.
I believe any coupling between areas of the brain must be along axons.

Yea I arrived at a similar conclusion, the near-field shouldn't be responsible for any coupling. I then found the Soliton Model of Neuroscience and am intrigued. So intrigued that I must pursue my PhD and the University of Copenhagen working on the model, experimentally, computationally, and theoretically.

 

[Baluncore]

Fascinating, and a good move. Maybe it is time to subvert the dominant paradigm.

So the solitons are guided by the axon's differential acoustic velocity factors.
I would expect the solitons to travel on the inside surface of the myelin sheath.

Maybe the gaps every few millimetres, are inline amplifiers?

Solitons are often self-sharpening due to non-linear duct characteristics. That would be a very useful property in an axon.