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A Mathematical Model of a Neuron with Synapses based on Physiology
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- Tokyo Institute of Technology, Precision and Intelligence Laboratory
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- Date:
- Received 21 March 2008 02:25 UTC; Posted 26 March 2008
- Subjects:
- Neuroscience
- Abstract:
The neuron, when considered as a signal processing device, itsinputs are the frequency of pulses received at the synapses, and its output is the frequency of action potentials generated- in essence, a neuron is a pulse frequency signal processing device. In comparison, electrical devices use either digital or analog signals for communication or processing, and the mathematics behind these subjects is well understood. However, in regards to pulse frequency processing devices, there has not yet been a clear and persuasive mathematical model to describe the functions of neurons. It goes without saying that such a model is very important, not only for understanding neuron and neural system behavior, but also for undeveloped potential applications in industry. This paper proposes a method for obtaining the mathematical relationship between the input and output signals of a neuron based on physiological facts. The proposed method focuses on the currents across the postsynaptic membrane of each synapse, and the key is to recognize that the net charge across the whole membrane of a neuron over each action potential cycle must equal to zero. By analyzing the relationship between the input of a synapse and the currents across the postsynaptic membranes, a dynamic pulse frequency model of the neuron can be obtained. Here, we show that the transfer function of a neuron depends on the function of thepostsynaptic current of each synapse in resting state, which can be found by detecting the postsynaptic current when a pulse is received at the synapse. The transfer function of a typical neuron generally includes addition and subtraction of feedthrough terms and/or first order lag functions. To focus on the most basic characteristics of a neuron, accommodation, adaptation, learning, etc. are not discussed in this paper.
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4 comments
Dear Danko Georqiev
Thank you very much!
But the total charges would not be changed by an interanal organ. For example, if a mitochondria released a Ca2+, the mitochondria will become 2-, the total charges is not changed. This process also can be explained using a 2 capacitors circuit.Dear Xiaolin Zhang,
I don’t understand well your logic. Remember that there are two different types of processes – passive and active. Active means that for release of energy the resistance of membranes and conductivity for various ion types is changed.
Keep in mind that during action potential only a millionth fraction of total ions move to create the membrane potential according to Goldman’s equation. So even a small deviation of ionic concentrations has huge imapct on neuronal physiology.
Also pH outside neurons is say ~7.4, this means an excess of negative charges (OH-, Cl-) according to Lewis acid-base theory, inside neurons the pH is ~ 6 or lower, so there is an excess of positive charges (H+, K+).
You cannot sum these charges to zero.
You said “the net charge across the whole membrane of a neuron over each action potential cycle must equal to zero”.
If you mean the net charge as sum of all + and – charges inside and outside neuron, then they will not give zero!
If you mean that the charge flux across membrane is zero, this is already clearly false.
I now understand that you want to sum the charges to zero, but due to energy production in neurons there is TOTAL production of H+ charges, which means the pH is always going down.
The only process that keep the balance is spending some energy to export the H+ out of neuron, and then clear from the body through the kidneys, our exhale CO2 through the lungs.
In any case the pH control is slower process compared to every action potential, and for each action potential the pH inside the neuron, and the pH outside the neuron matters.
The total charge cannot be summed up to zero, without calculation of the extracellular volume and the pH of it to get how many negative charges you have outside, and the volume of the neuron plus the pH in the cytosol to get how many positive charges you have inside.
—-NOTE—
Actually I don’t understand why total charge matters at all for neuronal function. What matters is the conductivity for given type of ions of the membrane.
Case [1]
Suppose there is 100 K+ and 1 Na+ ion inside and 100 Na+ and 1 K+ outside. And let us have K channel open.
You have 101 + inside and 101 + outsideThe K+ ions will go out, and you will have hyperpolarization.
Case [2]
Suppose there is 1 K+ and 100 Na+ ion inside and 1 Na+ and 100 K+ outside. And let us have K channel open.
You have 101 + inside and 101 + outsideThe K+ will go in opposite direction.
I suggest you read the basic theory in the Goldman-Hodgkin-Katz equations, and consider my comments above. Net charges have NO MEANING by itself.
In neuroscience matters WHAT TYPE of + or – charge you have, is it Na+, K+, Ca2+ etc, then look the Goldman equation, and then you will see in which direction the ions will flow knowing the resting potentials for the different ion types.
If the process is active, then everything is more complicated, e.g Hodgkin-Huxley equation, etc.
Dear Danko Georgiev,
Thank you again for the great comments!
Here are some ideas I have been thinking about:
1. even if ER or the mitochondria releases Ca2+ or H+, there will be no change in the membrane potential since there is no net change in charge within the membrane. Therefore, even as you say, that pH decreases due to H+ production, the net charge within the cell membrane will still be the same.
2. if we consider that the resting membrane potential to be constant (across multiple action potentials), then there cannot be a net change in charge within the cell membrane. Otherwise, the membrane potential will be different if there was a net change in charge.
If you are interested in an equation for the resting membrane potential of a cell, I have a paper titled “A new equivalent circuit different from the Hodgkin-Huxley model, and an equation for the resting membrane potential of a cell” in Artif Life Robotics(2002)6:140-148
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- This document is licensed to the public under the Creative Commons Attribution 3.0 License
- How to cite this document:
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Zhang, Xiaolin. A Mathematical Model of a Neuron with Synapses based on Physiology. Available from Nature Precedings <http://hdl.handle.net/10101/npre.2008.1703.1> (2008)
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Danko Georgiev on 27 April 2008 03:40 UTC
Dear Xiaolin Zhang, your work seems interesting, and I think it is a good starting point for creating a valid mathematical description of neuronal function. However there is one sereious problem with your statement
“and the key is to recognize that the net charge across the whole membrane of a neuron over each action potential cycle must equal to zero”.
This is physiologically NOT true. Why? I guess you consider a 2 compartment model – so + charges flow in, then they should flow out, etc. But indeed neuron is described better by 3 compartment model! There are internal stores such as endoplasmatic reticulum or mitochondria, which for example can store Ca2+ ions. Or alternatively they can release Ca2+ ions amplifying synaptic input in RyR and IP3R manner.
Surely at some time the net charge across the whole membrane will be zero because the internal store are limited in capacity, but this is not necessary to be the time for one action potential. Suppose that there is high frequency of action potentials. In this case there is not enough time for all the Ca2+ to be extruded out of the neuron, so mitochondria can store a portion of the Ca2+. When the neuron is not activated, the frequency of spikes is low, and in this relatively “resting” state, the mitochondria (or ER) can release part of the Ca2+ to be pumped out of the neuron, if these internal stores, were already (over)loaded.
I hope this will give you a hint how to improve your model, and know that the key statement given by you is wrong.
Maybe interesting for you will be this review paper “The nervous principle: active versus passive electric processes in neurons”
http://www.sciuni.com/neuroscience.html