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MBW:Mathematical neuroscience: from neurons to circuits to systems

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This is a summary of Mathematical neuroscience: from neurons to circuits to systems by B. Gutkin, D. Pinto and B. Ermentrout, Journal of Physiology – Paris, 209-219 (2003). All information comes from this source. Article reviewed by Nick Levine.

Executive Summary

Mathematics has played a crucial role in modeling neurons for years and has greatly increased our awareness of neural function. The following article summary will demonstrate the application of this modeling using three examples. First, by starting with a basic electrical circuit model, one can use phaseplane methods to simplify the circuit to a scalar system. Analyzing this system allows us to justify single neuron response to noisy stimuli. Further simplifying the circuit model leads to another model known as the activity model. This model analyzes excitatory and inhibitory neurons in networks and is used to explain the sensory response of rodents to stimuli. Lastly, we apply bifurcation theory to large spatial networks to describe patterns seen during visual hallucinations. [1]

Overview

Mathematics Used

Most of the mathematics involved in this model is differential equations describing the flow of current in different circuits. These differential equations correlate voltage, current, conductance, and other electric components with the constituents defined within the circuits. By observing changes in the current within these circuit elements given changes in time and frequency, the results can be extrapolated to describe biologically observable phenomena. (MR)

Type of Model

The original model describing neural systems as a series of circuits was the Hodgkin-Huxley model. Their original work described cells, the components of which each had a corresponding circuit element. In this paper the same basic principles apply, but the model is adjusted to think of neural systems as a series of planes. This creates less complex mathematics capable of modeling the behavior of neural systems under different circumstances. For another application of the Hodgkin-Huxley Model, see Period Doubling and other Phenomena in the Hodgkin-Huxley equations. (MR)

Biological System Studied

The original Hodgkin-Huxley model describes mostly cells; the cell components each have a corresponding circuit element. For example, the phospholipid bilayer of a cell is represented by a capacitor. Gutkin et al. adapt the model to describe neurons and neural systems. The adjusted model used in this paper, in addition to describing cells, can describe specific responses of different animals in different situations, such as describing information storage and explaining hallucinations. (MR)

Example 1: Equivalent Circuit

The circuit show below has become standard in describing neural function. Currents are divided into active and passive: passive represented by linear circuit elements and active require more complex analysis because of their dependency on voltage and/or time.

Circuit.bmp

  • Passive currents – We can make 3 observations about passive currents:
  1. Neurons have a resting potential of approximately -65 mV. This is due to the fact that neural membranes are semi-permeable, they maintain a concentration gradient across the membrane and the ions involved carry an electrical charge. In the circuit, this resting potential is the electromotive force or battery.
  2. The membrane behaves much like a resistor. As we change induced current, voltage changes resulting in a linear relationship. We represent this as gL, a single leak conductance for all passive ionic currents.
  3. Neural membranes also act as a RC circuit in that they behave like a linear resistor in parallel with a capacitor (voltage changes due to current injection follows an exponential relationship). In the neuron this happens due to capacitance buildup on the lipid bi-layer and the current flow through the protein channel.
  • Active currents – The passive currents described above cannot wholly describe a neurons response because some non-linearity exists. Hodgkin and Huxley[2]introduce a gating model that describes this behavior. Protein channels are also charged which open and close to allow current flow in a time dependent manner. In the first order kinetics equation below the probability of being open for current flow is represented by n. Additionally two types of gating particles are modeled, activation gates (m) and inactivation gates (h).

Equation 1.bmp

Taking the circuit above and applying Kirchoff’s First Law (the sum of all current flowing toward a junction is zero) the equations above are derived. These are called current-balance equations. Not only does this model basic neural behavior, but the equations were also used by Hodgkin and Huxley as a basis to describe voltage spikes that initiate signaling between neurons.

To learn more about Hodgkin and Huxley and their work, please refer to APPM4390:Hodgkin-Huxley

Another application of these equations is known as the dynamic clamp. This model uses real neurons attached to simulated ionic currents in which an electrode can either add or subtract new electrical channels and study the subsequent neural behavior. This has been used in lobster digestive systems, mammalian brains, and simulated neuron circuits modeled on programmable microchips. More recent studies have focused on neural response to brain rhythms during sleep.

To read more about noise in the nervous system please see APPM4390:Noise in the Nervous System

Morris and Lecar: 2 dimensional model

The Hodgkin and Huxley model was very complex in the sense that it possessed 4 dimensions. In order to simplify the math further, Morris and Lecar[3] developed a 3 dimensional model. This includes a fast calcium channel, slow potassium channel, and a passive leak channel. Because the calcium channel is extremely fast it is considered to be instantaneous and the model is considered two dimensional. The current balance equation for this two dimensional model is below. This also allows phaseplane analysis to be used to compare one variable as a function of another.

Equations.bmp Graph.bmp

An example of phaseplane analysis is shown above. This models a membrane generating an action potential. Point R represents the resting potential. Point T represents the threshold potential. These points also represent the steady states of the model. From the phaseplane diagram we see that injecting currents results in a voltage increase which will in turn raises the voltage nullcline. If the increase is large enough, the rest and threshold voltages converge and produce a limiting cycle. The shape of the cells trajectory does not change no matter how fast the cell fires allowing the application of bifurcation theory. This yields one positive eigenvalue that passes through zero. This saddle node is the basis for treating the loop as an invariant. To find the center of the loop, we use the fact that saddle node bifurcations follow the equation below:

Bifur.bmp

where x describes the dynamics, p, q and I are system parameters. By wrapping x on a unit circle and substituting x=tan(θ/2) results in (known as the theta-neuron model):

Theta.bmp

Understanding the model above has resulted in some significant findings. Such examples are that action potentials are all or none events, constant current injections result in continuous generation of action potentials, and repetitive firings are infrequent.

How does a neuron code information about the world? Is it through the average frequency of spikes or the precise timing of the spikes? The theta-neuron model above can explain both.

With noisy voltage but a constant current the neuron behaves like a non-linear renewal process. A successive spike depends on the previous spike and since the spikes are voltage dependent, uncertainty builds. In contrast, if a fluctuating current is introduced the spike times do not depend on one another. The current pushes the voltage beyond the threshold voltage the cycle is reset at the resting potential.

An in-depth summary of the Morris-Lecar model can be found at MBW:The Morris-Lecar Model for Excitable Systems.

Activity Models:Deriving the equations

In order to provide a more functional description of neural activity an "activity model" is used. Where the current balance equations above modeled a single neuron, it is strongly desired to model more generalized activity levels in single neuron or groups of neurons. Most often used to study the interactions between neurons, the activity model is even more simplified than the current balance equations.

The basic equation for the activity model focuses on voltage, firing rate, and synaptic drive. Synaptic drive is voltage changes due to other neuron firings in the vicinity of a specific neuron or network. This voltage change can be positive or negative depending on the state of the synapse, mainly if it is excitatory or inhibitory and is depicted as an alpha function[4] that leads to the equation below.

Equation3.bmp

Because the average firing rate depends on the average voltage, we define F(t)=F(V(t)) where F(t) represents the average firing rate. We then split the population into excitatory and inhibitory population and this results in the average excitatory population voltage given by:

Equation4.bmp

where ie represents the strength of synapse from inhibitory to excitatory (similarly, ee represents the strength of synapse from excitatory to excitatory). This model can be expressed differently if we define the synaptic drive of each population to be:

Equation5.bmp

We can alternatively define the excitatory population based on the equation above to be:

Equation6.bmp

Finally if we assume that the alpha function can be modeled using exponential decay we can differentiate and find that:

Equation7.bmp

This activity model is more easily correlated with biological data to model systems and is much more applicable in laboratory research.

Example 2: Application of the Activity Model: Whiskers and Barrels

A whisker barrel is part of neocortex that is the central receiving facility for all neural stimulation when the whisker on a rodents face is stimulated. It contains both excitatory and inhibitory synapses and the goal is to apply the activity model above to understand how the whisker barrel operates. This model is pictured below.

Whisker.bmp

In the lab, average firing rates are grouped into each respective neuronal population (excitatory and inhibitory) and the thalamus is the third data point. Here we model the thalamus as an excitatory neuron population only thus allowing it to be directly inputted into the model. As described above for activity models, F represents the firing rate function and the exact shape of this function is determined experimentally. In its application, the model shows that the faster the thalamic activity event occurs, the larger the response received from the system. If the same thalamic activity is inputted but spread out over a larger period of time, the networks response is significantly smaller. This is seen in the phaseplane analysis below.[5]

Graph2.bmp

Example 3: Cave Paintings and Structure of the Cortex

Anthropologists have long tried to link early Paleolithic rock paintings and patterns seen during hallucinations. Similar patterns result from eyeball pressure and flickering light. These are called phosphenes and are broken into four types referred to as tunnels and funnels, spirals, lattices, and cobwebs.

The goal of this example is to apply the activity model to a network that is coupled in a spatially dependent manner. Unlike the whisker barrel, the visual cortex is organized in columns and neurons respond to spatial positions of the retina, inputs from left to right, and oriented lines. Taking a more general form of the excitatory population equation used in the whisker model results in the equation below, which allows for the coupling of many neurons.

Equation8.bmp

We then linearize the equation above and investigate the stability. Very structured networks like the visual cortex which possess instabilities result in transcritical or pitchfork bifurcations. We can exploit this instability through the use of external input by increasing the excitability of the network. It is important to note the light imposed on the retina results in a map on the visual cortex. Similarly, a pattern of activation in the cortex results in a virtual pattern on the retina. This map idea and the fact that we can create stability loss via external input leads directly to the abstract designs of the early cave paintings. Ermentrout and Cowan[6] used group theory and exploited the fact that the visual cortex responds to spatial position to dramatically simplify the equation above. The following arises:

Equation9.bmp

where a and b are model dependent constants and A1 (A2) is the amplitude of horizontal (vertical) stripes. If A1≠ 0, A2=0 (A1=0, A2 ≠0) the horizontal (vertical) strips occur and if A1=A2 spots occur.

Pictured below are bifurcation patterns.

Lastpic.bmp

Further Reading

Expanding on this model, and in particular its applications in describing visual hallucinations, Billock and Tsou describe three different experiments carried out to better quantify the types of stimuli contributing to visual hallucinations.[7] The first experiment dealt with forcing hallucinations by biasing towards certain shapes, such as spirals or fans, to see if certain physical stimuli can induce cortical activity associated with different stimuli. A second experiment following a similar procedure involved icons flashing certain colors to induce hallucinations of other colors. Finally, by filtering images of noise (such as television “snow”), certain patterns can be induced. (MR)

External Links

  1. B.Ermentrout, B. Gutkin, D. Pinto Mathematical neuroscience: from neurons to circuits to systems. Journal of Physiology, Paris. 209-219 (2003).
  2. A.L. Hodgkin, A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J.Phys.(London) 117 (1952) 500–544.
  3. J.Rinzel, G.B. Ermentrout, Analysis of neural excitability and oscillations, in: C.Koch, I.Segev (Eds.), Methods in Neuronal Modeling, MIT Press, Cambridge, MA, 1998.
  4. W.Rall, A.Agmon-S nir, Cable theory for dendritic neurons, in: C.Koch, I. Segev (Eds.), Methods in Neuronal Modeling, MIT Press, Cambridge, MA, 1998, pp.27–93.
  5. D.J. Pinto, J.C. Brumberg, D.J. Simons, Circuit dynamics and coding strategies in rodent somatosensory cortex, J.Neurophy siol. 83 (2000) 1158–1166..
  6. G.B. Ermentrout, J.D. Cowan, A mathematical theory of visual hallucinations, Kybernetic 34 (1979) 137–150..
  7. Vincent A. Billock and Brian H. Tsou, Neural interactions between flicker-induced self-organized visual hallucinations and physical stimuli. PNAS May 15, 2007, vol. 104, no. 20 (8490-8495).