Computation in networks

In this article, we will discuss some of the principles behind models of biological information processing. As usual, we therefore use tools at the interface between science, mathematics and computer science. We want to find the right abstraction of the wealth of biological detail that is available; the right design to guide us in the development of our modeling environment. The details of how proteins interact, how genes interact and how neural modules interact to generate the high level outputs we find both interesting and useful are known to some degree but it is quite difficult to use this detailed information to build models of high level function. In all models of this type, we are asking high level questions and wondering how we might create a model that gives us some insight. All such models have built in assumptions and we must train ourselves to think abut these carefully. We must question the abstractions of the messiness of reality that led to the model and be prepared to adjust the modeling process if the world as experienced is different from what the model leads them to expect. There are three primary sources of error when we build models. First, there is the error we make when we abstract from reality; we make choices about which things we are measuring are important. We make further choices about how these things relate to one another. Perhaps we model this with mathematics, diagrams, words etc; whatever we choose to do, there is error we make. This is called Model error. Then, we make error when we use computational tools to solve our abstract models which is called Truncation error. Finally, since we can not store numbers exactly in any computer system, there is always a loss of accuracy because of this. This is called Round Off error. All three of these errors are always present and so the question is how do we know the solutions our models suggest relate to the real world? We must take the modeling results and go back to original data to make sure the model has relevance.

With all this said, let’s start looking at the fundamental building blocks of information transmission in a living neural system. The neurotransmitters and other signaling molecules to accomplish coordination of effort between individual cells were laid down probably in Cambrian times, so these pathways are very old. However, control of a multi-cellular organism such as ourselves does not necessarily require a neural architecture like ours. Recent work in the Ctenophore genome has shown us a new neural architecture for control (see (Moroz et al. 2014), (Callaway 2014), (Staff 2013) and (Ryan et al. 2013)). Also, single cells can be controlled in what appears to be the same way a multi-cellular organism can be as is seen in the video (Nautilus 2014). This implies our framework for understanding how to build useful neural architectures for our goals (understanding cognition, drug treatment, autonomous movement and so forth) does not have to be tied to the architectures we see in invertebrate and vertebrate animals. This implies we are not limited to ideas from invertebrate and vertebrate physiology to solve problems in computational cognition modeling.

We start with a simple cell and ion movement in and out of the cell ((Johnson and Wu 1995), (Weiss 1996a) and (Weiss 1996b)). These references provide even more detail and you should feel free to look at these books. However, the amount of detail in them can be overwhelming, so we offer a short version with just enough detail for our mathematical/ biological engineer and computer scientist audience.

An ion c can move across a membrane due to several forces. First, let’s talk about what concentration of a molecule means. For any molecule b, the concentration of the ion is denoted by the symbol [b] and is measured in \(\frac {molecules}{liter}\). Now, we hardly ever measure concentration in molecules per unit volume; instead we use the fact that there are N _A =6.02 × 10²³ molecules in a Mole and usually measure concentration in the units \(\frac {Moles}{cm^{3}} \: = M\) where for simplicity, the symbol M denotes the concentration in Moles per c m ³. The special number N _A is called Avogadro’s Number. The force that arises from the rate of change of the concentration of molecule b acts on the molecules in the membrane to help move them across. The amount of molecules that move across per unit area due to this force is labeled the diffusion flux as flux is defined to a rate of transfer \(\left (\frac {something}{second}\right)\) per unit area.

There are several basic laws to consider. Ficke’s Law of Diffusion is an empirical law which says the rate of change of the concentration of molecule b is proportional to the diffusion flux and is written in mathematical form as \(J_{\textit {diff}} = - D \: \frac {\partial \: [b]}{\partial x}\) where J _diff is diffusion flux which has units of \(\frac {molecules}{cm^{2}-second}\), D is the diffusion coefficient which has units of \(\frac {cm^{2}}{second}\) and [b] is the concentration of molecule b which has units of \(\frac {molecules}{cm^{3}}\). The minus sign implies that flow is from high to low concentration; hence diffusion takes place down the concentration gradient. Note that D is the proportionality constant in this law.

Ohm’s Law of Drift relates the electrical field due to an charged molecule, i.e. an ion, c, across a membrane to the drift of the ion across the membrane where drift is the amount of ions that moves across the membrane per unit area. In mathematical form J _drift =− ∂ _el E where it is important to define our variables and units very carefully. We have J _drift is the drift of the ion which has units of \(\frac {molecules}{cm^{2}-second}\) and ∂ _el is electrical conductivity which has units of \(\frac {molecules}{volt-cm-second}\). Now the valence of ion c is the charge on the ion as an integer; i.e. the valence of C l ⁻ is −1 and the valence of C a ⁺² is +2. We let the valence of the ion c be denoted by z. It is possible to derive the following relation between concentration [c] and the electrical conductivity ∂ _el :∂ _el =μ z [c] where dimensional analysis shows us that the proportionality constant μ, called the mobility of ion c, has units \(\frac {cm^{2}}{volt-second}\). Hence, we can rewrite Ohm’s Law of Drift as \(J_{\textit {drift}} = - \mu z [c] \: \frac {\partial V}{\partial x}\). We see that the drift of charged particles goes against the electrical gradient.

There is also a relation between the diffusion coefficient D and the mobility μ of an ion which is called Einstein’s Relation. It says \(D = \frac {\kappa T}{q} \: \mu \) where κ is Boltzmann’s constant which is \(1.38 \: 10^{-23} \frac {joule}{\circ K}\), T is the temperature in degrees Kelvin and q is the charge of the ion c which has units of coulombs. We note electrical work has units of coulombs - volts. Hence, we see \(\frac {\kappa T}{q} \: \mu \) has units \(\frac {volt-coulomb}{\circ K} \: \frac {\circ K}{coulombs} \: \frac {cm^{2}}{volt-second} \: = \: \frac {cm^{2}}{sec}\) which reduces to the units of D. Further, we see that Einstein’s Law says that diffusion and drift processes are additive because Ohm’s Law of Drift says J _drift is proportional to μ which by Einstein’s Law is proportional to D and hence J _diff .

The membrane capacitance of a typical cell is one micro fahrad per unit area. Typically, we use F to denote the unit fahrads and the unit of area is c m ². Since the inside and outside of the cell are separated by a biological membrane of the type, we can ask what if one side of the cell had more or less ions than the other? These uncompensated ions would produce a voltage difference across the membrane because charge is capacitance times voltage (q = c V). Hence, if we wanted to produce a 100 mV potential difference across the membrane, we can compute how many uncompensated ions, δ[c] would be needed: \(\delta [c] = \frac {10^{-6}F}{cm^{2}} \: \times \:.1 V = \: 10^{-7} \: \frac {coulombs}{cm^{2}}\). This is a typical voltage difference across a biological membrane in an excitable nerve cell. For a typical cell the total number of ions inside the cell for a.5M solution is ≈10¹⁶ ions. We can show the number of uncompensated ions per cell to get this voltage difference is 4.95 × 10⁷ ions which is only ≈ 10⁻⁷ % of the total. Hence, the voltage differences relevant to excitable nerve cell are achieved by moving a tiny fraction of the ions available across the membrane.

where we let [ c] _out be the concentration of c outside the cell and [ c] _in be the concentration of c inside. This important equation is called the Nernst equation and is an explicit expression for the equilibrium potential of an ion species in terms of its concentrations inside and outside of the cell membrane. For example, in frog muscle, K ⁺ has an interior concentration of 124.0 mM and an outer concentration of 2.25 mM leading to an equilibrium voltage of −101.52 mV.

The electrical potential across the membrane is determined by how well molecules get through the membrane (its permeability) and the concentration gradients for the ions of interest. If there are multiple ions involved, what determines the resting potential? Recall Ohm’s Law for a simple circuit: the current across a resistor is the voltage across the resistor divided by the resistance; in familiar terms \(I \: = \: \frac {V}{R}\) using time honored symbols for current, voltage and resistance. It is easy to see how this idea fits here: there is resistance to the movement of an ion through the membrane. Since \(I \: = \: \frac {1}{R} \: V\), we see that the ion current through the membrane is proportional to the resistance to that flow. The term \(\frac {1}{R}\) seems to be a nice measure of how well ions flow or conduct through the membrane. We will call \(\frac {1}{R}\) the conductance of the ion through the membrane. Conductance is generally denoted by the symbol g. Clearly, the resistance and conductance associated to a given ion are things that will require very complex modeling even if they are pretty straightforward concepts, but for now, for an ion c, we will use I _c =g _c (V _m − E _c ) where E _c is the equilibrium voltage for the ion c that comes from the Nernst Equation, I _c is the ionic current for ion c and g _c is the conductance. Finally, we denote the voltage across the membrane to be V _m . Note the difference between the membrane voltage and the equilibrium ion voltage provides the electromotive force or emf that drives the ion. Consider Figure 1. We are thinking of a patch of membrane as a parallel circuit with one branch for each of the three ions K ⁺, N a ⁺ and C l ⁻ and a branch for the capacitance of the membrane. The branch for C l ⁻ in the figure is labeled as a leakage current with conductance g _L and Nernst voltage E _L . Later in the Hodgkin - Huxley model, we will let this leakage current include things other than the chlorine currents, but for now, it is just what we get from the chlorine ion. We think of this patch of membrane as having a voltage difference of V _m across it. In general, there will current that flows through each branch. These currents are i _K , i _Na , i _L =i _Cl and i _c where I _c denotes the capacitative current. For right now, we will assume all of these conductances are constant. Each of our ionic currents have the form i _ion =g _c (V _m − E _c ) where V _m , as mentioned, is the actual membrane voltage, c denotes our ion, g _c is the conductance associated with ion c and E _c is the Nernst equilibrium voltage). Hence for three ions, potassium (K ⁺), sodium (N a ⁺) and chlorine (C l ⁻), we have i _K =g _K (V _m − E _K ), i _Na =g _Na (V _m − E _Na ) and i _Cl =g _Cl (V _m − E _Cl ). There is also a capacitative current. We know the voltage drop across the capacitor C _m is given by \(\frac {q_{m}}{V_{m}}\); hence, the charge across the capacitor is C _m V _m implying the capacitative current is \(i_{m} = C_{m} \frac {{dV}_{m}}{dt}\) At steady state, i _m is zero and the ionic currents must sum to zero giving i _K + i _Na + i _Cl =0. Hence, 0=g _K (V _m − E _K ) + g _Na (V _m − E _Na ) + g _Cl (V _m − E _Cl ) leading to a Nernst Voltage equation for the equilibrium membrane voltage V _m of a membrane permeable to several ions:

$$ \begin{aligned} V_{m} =& \frac{g_{K}}{g_{K} + g_{Na} + g_{Cl}} \: E_{K} + \frac{g_{Na}}{g_{K} + g_{Na} + g_{Cl}} \: E_{Na}\\ &+ \frac{g_{Cl}}{g_{K} + g_{Na} + g_{Cl}} \: E_{Cl} \end{aligned} $$

(6)

Hence, if we are given the needed conductance ratios, we can compute the membrane voltage at equilibrium for multiple ions.

Our abstract cell is a spherical ball which encloses a fluid called cytoplasm. The surface of the ball is actually a membrane with an inner and outer part. Outside the cell there is a solution called the extracellular fluid. Both the cytoplasm and extracellular fluid contain many molecules, polypeptides and proteins disassociated into ions as well as sequestered into storage units. We are now interested in what this biological membrane is and how we can model the flow of ionized species through it. This modeling is difficult because some of the ions we are interested in can diffuse or drift across the membrane and others must be allowed entry through specialized holes in the membrane called gates or even escorted, i.e. transported or pumped, through the membrane by specialized helper molecules.

In Figure 2, we see a schematic of a typical voltage gate. Note that the inside of the gate shows a structure which can be in an open or closed position. The outside of the gate has a variety of molecules with sugar residues which physically extend into the extracellular fluid and carry negative charges on their tips. At the outer edge of the gate, you see a narrowing of the channel opening which is called the selectivity filter. Proteins can take on very complex three dimensional shapes. Often, their actual physical shape can switch from one form to another due to some external signal such as voltage. This is called a conformational change. In a voltage gate, the molecule which can block the inner throat of the gate moves from its blocking position to its open position due to such a conformational change. In fact, this molecule can also be in between open and closed as well. The voltage gated channel is actually a protein macromolecule which is inserted into an opening in the membrane called a pore. This macromolecule is quite big (1800 - 4000 amino acids) with one or more polypeptide chains and 100’s of sugar residues hang off the extracellular face. When open, the channel is a water filled pore with a fairly large inner diameter which would allow the passage of many things except that there is one narrow stretch of the channel called a selectivity filter which inhibits access. The inside of the pore is lined with hydrophilic amino acids which therefore like being near the water in the pore and the outside of the pore is lined with hydrophobic amino acids which therefore dislike water contact. These therefore lie next to the lipid bilayer. Ion concentration gradients can be maintained by selective permeabilities of the membrane to various ions. Most membranes are permeable to K ⁺, maybe C l ⁻ and much less permeable to N a ⁺ and C a ⁺². This type of passage of ions through the membrane requires no energy and so it is called the passive distribution of the ions. There are also many pumps within a cell that move substances in or out of a cell with or against a concentration gradient, but we won’t discuss them here.

There are specialized cells in most living creatures called neurons which are adapted for generating signals which are used for the transmission of sensory data, control of movement and cognition through mechanisms we don’t fully understand. However, a neuron is an example of what is called an excitable cell whose membrane is studded with many voltage gated sodium and potassium channels. The equilibrium voltage of the cell membrane in terms of the conductances of each ion is

$$\begin{aligned} {V^{m}_{0}} =& \frac{g_{K}}{g_{K} + g_{Na} + g_{Cl}} \: E_{K} + \frac{g_{Na}}{g_{K} + g_{Na} + g_{Cl}} \: E_{Na}\\ &+ \frac{g_{Cl}}{g_{K} + g_{Na} + g_{Cl}} \: E_{Cl} \end{aligned} $$

The conductance model allows us to understand this sudden increase in voltage across the membrane in terms of either sodium to potassium conductance ratio shifts. In an excitable cell, under certain circumstances, the rest potential across the membrane can be stimulated in the right manner to cause a rapid rise in the equilibrium potential of the cell, followed by a sudden drop below the equilibrium voltage and then ended by a slow increase back up to the rest potential. The shape of this wave form is very characteristic and is shown in Figure 3. This type of wave form is called an action potential and is a fundamental characteristic of excitable cells. In the figure, we draw the voltage across the membrane and simultaneously we draw the conductance curves for the sodium and potassium ions. Since conductance is reciprocal resistance, a spike in sodium conductance, for example, is proportional to a spike in sodium ion current. So in the figure we see that sodium current spikes first and potassium second.

These gates transition from resting to open when the membrane depolarizes due to an incoming voltage pulse. In detail, the probability that the gate opens increases upon membrane depolarization. However, the probability that the gate transitions from open to closed is NOT voltage dependent. Hence, no matter what the membrane voltage, once a gate opens, there is a fixed probability it will close again.

Hence, an action potential can be described as follows: when the cell membrane is sufficiently depolarized, there is an explosive increase in the opening of the sodium gates which causes a huge influx on sodium ions which produces a short lived rapid increase in the voltage across the membrane followed by a rapid return to the rest voltage with a typical overshoot phase which temporarily keeps the cell membrane hyperpolarized.

Next, let’s look at how these depolarizing pulses come about.

We begin with a simple model of a biological cell. We can think of a cell as having an input line (this models the dendritic tree), a cell body (this models the soma) and an output line (this models the axon). We could model all these elements with cables – thin ones for the dendrite and axon and a fat one for the soma. To make our model useful, we need to understand how current injected into the dendritic cable propagates a change in the membrane voltage to the soma and then out across the axon. In a uniform isolated cell, the potential difference across the membrane depends on where you are on the cell surface. Now we wish to find a way to model V ^m as a function of the distance downstream from site at which a current is injected into the cable and also in terms of the the time elapsed since current injection. This model will be called the Core Conductor Model.

Building the core conductor model

Now let’s look at a slice of the model between positions z and z + Δ z. In Figure 5, we see an abstraction of this. The vertical lines represent the circular cross sections through the cable at the positions z and z+Δ z. We cut through the outer shell first giving two horizontal lines labeled outer membrane at the bottom of the figure. Then as we move upward, we encounter the outer conductor. We then move through the inner cable which has a bottom membrane, an inner conductor and a top membrane. Finally, we enter the outer conductor again and exit through the outer membrane. At each position z, there are values of I _i , I _o , V _i , V _o , K _m and V _m that are shown. Between z and z+Δ z, we thus have a cylinder of inner and outer cable of length Δ z. From the Δ z slice in Figure 5, we can abstract an electrical network model and using the standard Kirchoff laws of voltage and current balance at the relevant nodes, we find

$$\begin{array}{@{}rcl@{}} & &\frac{I_{i}(z^{*},t) - I_{i}(z,t)}{\Delta z} = -K_{m}(z,t), \quad \frac{I_{o}(z^{*},t) - I_{o}(z,t)}{\Delta z} \\&&\hspace*{71pt}= K_{m}(z,t) - K_{e}(z,t)\\ & &\frac{V_{i}(z^{*},t) - V_{i}(z,t)}{\Delta z} = -r_{i} I_{i}(z^{*},t), \quad \frac{V_{o}(z^{*},t) - V_{o}(z,t)}{\Delta z}\\ &&\hspace*{78pt}= -r_{o} I_{o}(z^{*} z,t) \end{array} $$

where to save space we have let z ^∗ denote z+Δ z. Now the equations above apply for any choice of Δ z. Physically, we expect the voltages and currents we see here to be smooth differentiable functions of z and t. Hence, we expect that if we let Δ z go to zero, we will obtain the Core equations: we have V _m =V _i − V ₀ and

$$\begin{array}{*{20}l} \frac{\partial I_{i}}{\partial z} &= -K_{m}(z,t), \quad \quad \frac{\partial I_{o}}{\partial z}= K_{m}(z,t) - K_{e}(z,t) \end{array} $$

(8)

$$\begin{array}{*{20}l} \frac{\partial V_{i}}{\partial z} &= -r_{i} I_{i}(z,t), \quad \quad \frac{\partial V_{o}}{\partial z} = -r_{o} I_{o}(z,t) \end{array} $$

(9)

Note Equation 9 implies that \(\frac {\partial V_{m}}{\partial z} = \frac {\partial V_{i}}{\partial z} \: - \: \frac {\partial V_{0}}{\partial z}\). Then, again from Equation 9, we have \(\frac {\partial V_{m}}{\partial z} = -r_{i} I_{i} \: + \: r_{o} I_{o}\) Thus, using Equation 8, we obtain

$$\frac{\partial^{2} V_{m}}{\partial z^{2}} =-r_{i} \frac{\partial I_{i}}{\partial z} \: + \: r_{o} \frac{\partial I_{0}}{\partial z} = r_{i} K_{m} \: + \: r_{o} K_{m} \: - \: r_{o} K_{e} $$

Thus, the core equations imply that the membrane voltage satisfies the Core Conductor Equation

$$ \frac{\partial^{2} V_{m}}{\partial z^{2}} =(r_{i} \: + \: r_{o}) K_{m} \: - \: r_{o} K_{e} $$

(10)

The new constants τ _m and λ _c are very important to understanding how the solutions to this equation will behave. We call τ _m the time constant and λ _c the space constant of our cable. To understand the Time Constant, consider the ratio \(\frac {c_{m}}{g_{m}}\). Note the ratio has units of seconds and thus, we can interpret this constant as the time constant of the cable, τ _m . Note that τ _m is a constant whose value is independent of the size of the cell; hence it is a membrane property. We can show the time constant can thus be expressed as \(\frac {c_{m}}{g_{m}}\) also. Next consider the Space Constant. Look at the dimensional analysis of the term \(\frac {1}{(r_{i}+r_{o}) g_{m}}\) which has units of cm ². This is why the square root of the ratio functions as a length parameter. We can look at this more closely. Consider again our approximate model where we divide the cable up into pieces of length Δ z. If we do some standard biophysical analysis, we can show \(\lambda _{c} = \sqrt {\frac {a}{2 \rho _{i} G_{m}}}\). Now ρ _i and G _m are membrane constants independent of cell geometry. So we see that the space constant is proportional to the square root of the fiber radius. Note also that the space constant decreases as the fiber radius shrinks.

We now extend our simple dendritic cable model to what is called the ball and stick neuron model. This consists of an isopotential sphere to model the cell body or soma coupled to a single dendritic fiber input line. We model the soma as a simple parallel resistance/capacitance network and the dendrite as a finite length cable as previously discussed (see Figure 6). In Figure 6, you see the terms I ₀, the input current at the soma/dendrite junction starting at τ=0;I _D , the portion of the input current that enters the dendrite (effectively determined by the input conductance to the finite cable, G _D ); I _S , the portion of the input current that enters the soma (effectively determined by the soma conductance G _S ); and C _S , the soma membrane capacitance. We assume that the electrical properties of the soma and dendrite membrane are the same; this implies that the fundamental time and space constants of the soma and dendrite are given by the same constant (we will use our standard notation τ _M and λ _C as usual). For convenience, we now normalized the variables using the transformations

$$\tau = \frac{t}{\tau_{M}}, \: \: \lambda = \frac{z}{\lambda_{C}}, \: \: L = \frac{\ell}{\lambda_{C}}, \: \: \Longrightarrow \hat{v}_{m} = v_{m}(\lambda_{C} \lambda,\tau_{M} \tau). $$

It is then possible to show that with a reasonable zero-rate left end cap condition the appropriate boundary condition at λ=0 is given by

$$ \rho \: \frac{\partial \hat{v}_{m}}{\partial \lambda}(0,\tau) = \tanh(L)\left[\hat{v}_{m}(0,\tau) \: + \: \frac{\partial \hat{v}_{m}}{\partial \tau}(0,\tau) \right], $$

(14)

The calculation of the first Q coefficients is then handled as a linear algebra problem.

Note the differences between the solution we find by using separation of variables techniques where we assume the time and spatial parts of the solution are separated which forces us to use an infinite series approach. This also introduces numerical artifacts and so forth. The solution using transform tools shows a very different time and space dependence in the solution which means the decay of the response to the voltage impulse applied to the cable is actually different than a standard exponential decay. Still, a nice guiding principle is that the response drops roughly e ⁻¹ in magnitude every time and space constant we move away from the injection site. Also, please note that we make a large number of simplifying assumptions in order to arrive at these results.

We now discuss the Hodgkin - Huxley model for the generation of the action potential of an excitable nerve cell using the standard Ball and Stick model. This augmented cable equation can also be solved using the transform techniques but we will not do so here. As previously mentioned, there are many variables are needed to describe what is happening inside and outside the cellular membrane, and to some extent, inside the membrane for a standard cable model. In the standard core conductor model, the membrane is not modeled at all, but we now need to be more careful. A realistic description of how the membrane activity contributes to the membrane voltage must use models of ion flow which are controlled by gates in the membrane. A simple model of this sort is based on work that Hodgkin and Huxley ((Hodgkin and Huxley 1952), (Hodgkin 1952), (Hodgkin 1954)) performed in the 1950’s. We start by expanding the membrane model to handle potassium, sodium and an all purpose current, called leakage current, using a modification of our original simple electrical circuit model of the membrane. We will think of a gate in the membrane as having an intrinsic resistance and the cell membrane itself as having an intrinsic capacitance as shown in Figure 7. This is a picture of an idealized cell with a small portion of the membrane blown up into an idealized circuit: we see a small piece of the lipid membrane with an inserted gate. Thus, we expand the single branch of our old circuit model to multiple branches – one for each ion flow we wish to model. The ionic current consists of the portions due to potassium, K _K , sodium, K _Na and leakage K _L . The leakage current is due to all other sources of ion flow across the membrane which are not being explicitly modeled. This would include ion pumps; gates for other ions such as Calcium, Chlorine; neurotransmitter activated gates and so forth. We will assume that the leakage current is chosen so that there is no excitable neural activity at equilibrium. We know that conductance is reciprocal resistance, so our model will consist to a two branch circuit: one branch is contains a capacitor and the other, the conductance element. We will let c _m denote the membrane capacitance density per unit length (measured in \(\frac {fahrad}{cm}\)). Hence, in our membrane box which is Δ z wide, we see the value of capacitance should be c _m Δ z. Similarly, we let g _m be the conductance per unit length (measured in \(\frac {1}{ohm-cm}\)) for the membrane. The amount of conductance for the box element is thus g _m Δ z. In Figure 8, we illustrate our new membrane model. Since this is a resistance - capacitance parallel circuit, it is traditional to call this an RC membrane model. In Figure 8, the current going into the element is K _m (z,t)Δ z and we draw the rest voltage for the membrane as a battery of value \({V_{m}^{0}}\). We know thatthe membrane current, K _m , satisfies Equation 21:

$$ K_{m}(z,t) = g_{m} \: V_{m}(z,t) \: + \: c_{m} \frac{\partial V_{m}}{\partial t} $$

(21)

In terms of membrane current densities, all of the above details come from modeling the simple equation K _m =K _c + K _ion where K _m is the membrane current density, K _c is the current through the capacitative side of the circuit and K _ion is the current that flows through the side of the circuit that is modeled by the conductance term, g _m . We see that in this model \(K_{c} = c_{m} \frac {\partial V_{m}}{\partial t}\) and K _ion =g _m V _m . However, we can come up with a more realistic model of how the membrane activity contributes to the membrane voltage by adding models of ion flow controlled by gates in the membrane. Our models are based on work that Hodgkin and Huxley performed in the 1950’s.

Hodgkin and Huxley modeled the sodium and potassium gates as

where and are called activation variables and is an inactivation variable which all satisfy the first order Φ kinetics that tell us

where for each ion c

Further, the coefficient functions, α and β for each variable required data fits, such as , as functions of voltage. We will not list the rest here as we do not need that level of detail. Of course these data fits were obtained at a certain temperature and assumed values for all the other constants needed and so they need to be altered if the temperature and ionic concentrations change. Our model of the membrane dynamics here thus consists of the following differential equations:

We note that at equilibrium there is no current across the membrane. Hence, the sodium and potassium currents are zero and the activation and inactivation variables should achieve their steady state values which would be m _∞ , h _∞ and n _∞ computed at the equilibrium membrane potential which is here denoted by V ₀.

At this point, we see a general model of how to generate an action potential. We do not model the transmission of the action potential to its target neurons; suffice it to say, there are molecular mechanisms that send the pulse to its targets without change. At the interface to the target neuron called the synapse, the incoming voltage pulse generates a Ca ⁺² current which in turn generates discrete packets of neurotransmitter which are released for processing by the target neurons dendritic subsystem. Note the signal transduction pathway here: the axonal voltage pulse is converted into a discrete release of molecules which bind with the dendritic arbor and thereby shape the next axonal pulse in very nonlinear ways. We are also not interested here in how information might be encoded in collections of axonal spikes. Our interest at the moment is in the low level bones. In general, we separate the effects of these triggers into two classes: first messengers which are essentially voltage activated gates and second messengers which enter through the membrane and initiate a protein transcription event which can rebuild the actual hardware of the cell. An abstract analysis of such a trigger event is presented in (Peterson 2014a) in order to build trigger approximations. Here, we will primarily be interested in second messengers which are neurotransmitters even though the triggers can be more general. There is a vast literature on signaling in a biological neural system and we will point you to just a few relevant sources. We prefer papers and texts that help us understand an overarching theory of the signaling process. Most of these sources are actually fairly old but are full of the technical detail needed to understand the processes being studied. In particular, the use of mathematical points of view is not shied away from and newer references tend to downplay that point of view. Two basic and very useful texts on signaling are (Gerhart and Kirschner 1997) and (Wilkins 2002) which look at these ideas very theoretically. Two other very good resources for this material are (Hille 1992) and (Bray 1998). Bray’s paper is very important and is very useful when you try to understand signaling principles; again, the material is not really dated and it is worth a strong push towards assimilating its ideas. Remember that understanding signaling principles allows us to see how to build reasonable approximations. The paper (Sneppen et al. 2010) is really about protein networks but it has lots of good advice about how to approximate complicated biophysics and then glue together the models into systems. A good overview of how neurons work together to generate high level behavior (which we do not really understand) can be found in (Roberts et al. 2010) which will help you see Black’s points discussed in the next section again phrased in terms of what we currently understand. There is a lot of combinatorial complexity here too and modeling that is hard. Even 4 things taken 2 at a time at each node leads to incredible explosions of computation. A really interesting article on dealing with that is found in (Borisov et al. 2006) which is well worth a read. Another look at the notion of discrete synaptic states and resulting computational complexity is in (Montgomery and Madison 2004) and (Harris-Warrick and Johnson 2010). With all this said, for simplicity, let’s look now at the class of catecholamine neurotransmitters in the nervous system. We will focus on only a few types: DA, dopamine; NE, norepinephrine; and E, epinephrine and lump them together into the category called CAs because they all share a common core biochemical structure, the catechol group.

Clearly, dendritic–axonal software interaction should be mediated via pathways of both local scope objects) and global scope, using additional objects which could be modeled after hormones.

In our mind, although there is an architecture that specifies connection information between processing nodes, what really counts is how the computation is organized into overlapping computational modules. It is dangerous to think that these neural systems are organized hierarchically. In (Wagner 2014), there is a compelling discussion of hierarchical schemes which is targeted towards problems in homology, the study of how characteristics in different species are similar and could even have evolved from a common ancestor. Wagner says it best:

He goes on further to note that some cell types can be understood as different combinations of gene regulatory modules which is reminiscent of the kind of combinatorial machinations we see in the immune system to generate a response to an intruder ((Shastri et al. 2002) and (Rot and von Andrian 2004)) and in the nervous system ((Black 1991) and (Montgomery and Madison 2004)). Each IOMAP module must therefore be capable of a certain number of active states.

The inputs into the dendritic tree of an excitable nerve cell can be separated into first and second messenger classes. The first messenger group consists of Hodgkin-Huxley voltage dependent ion gates and the second messengers includes molecules which bind to the dendrite through some sort of interface and then trigger a series of secondary events inside the cytoplasm of the cell body which lead to protein transcription and the modification of the cell’s molecular machinery. We can connect a standard feedforward architecture of neurons as a chain as shown in Figure 9. Note that in Figure 9, the interaction between dendritic and axonal objects has not been made explicit. The usual simplistic approximation to this interaction transforms a summed collection of weighted inputs via a saturating transfer function with bounded output range. The dendritic and axonal interaction is modeled by a scalar W _ij as described above. It measures the intensity of the interaction between the input neuron i and the output neuron j as a simple scalar weight.

Chained architecture details

Let’s look at a chain of computational nodes such as neurons in more detail. If the architecture is feed forward we would have the Chain Feed Forward Network (CFFN). The chain consists of computational elements, generally referred to as neurons. This is because a very simple model of neural processing models the action potential spike with a sigmoid function which transitions rapidly from a binary 0 (no spike) to 1 (spike). This sigmoid is called a transfer function and since it can not exceed 1 as 1 is a horizontal asymptote, it is called a saturating transfer function also. This model is known as a lumped sum model of post-synaptic potential; now that we know about the ball - stick model of neural processing, we can see the lumped sum model is indeed simplistic. Each neuron thus processes a summed collection of weighted inputs via a saturating transfer function with bounded output range (i.e. [0,1]). The neurons whose outputs connect to a given target or postsynaptic neuron are called presynaptic neurons. Each presynaptic neuron has an output Y which is modified by the synaptic weight W _{p r e,p o s t} connecting the presynaptic neuron to the postsynaptic neuron. This gives a contribution W _{p r e,p o s t} Y to the input of the postsynaptic neuron. A typical transfer function could be modeled as \(\sigma (x,o,g) =.5 \left (1.0 + \tanh \left (\frac {x-o}{\phi (g)} \right) \right)\) where o is the offset which controls the location of the maximum response of the sigmoid and ϕ(g) is a function which shapes how fast the sigmoid rises from its minimum to its maximum value. The chained model then consists of a string of N neurons, labeled from 0 to N−1. Some of these neurons can accept external input and some have their outputs compared to external targets. We let \({\mathcal U} = \text { indices} ~i~ \text {where neuron}~ i~ \text {is an input}\) and \({\mathcal V} = \text {indices} ~i \text { where neuron}~ i~ \text {is an output}\). We will let n _I and n _O denote the size of and respectively. Note in a chain, it is also possible for an input neuron to be an output neuron; hence and need not be disjoint sets. Each neuron can be viewed as a postsynaptic neuron with a set of presynaptic neurons feeding into it: thus, each neuron i has associated with it a set of backward links which will be denoted by (i). Each node i also connects forward to other nodes and the set of forward connections is called \({\mathcal F}(i)\). The weight of the synaptic link connecting the presynaptic neuron i to the postsynaptic neuron j is then denoted by W _i→j although in general this should be an edge processing function. For a feedforward architecture, we would have j > i, however, as you can see in Figure 9, this is not true in more general chain architectures. The input of a typical postsynaptic neuron therefore requires summing over the backward link set of the postsynaptic neuron in the following way:

$$y^{post} = x + \sum\limits_{pre \: \in \: {\mathcal B}(post)} \: W_{pre \rightarrow post} \: Y^{pre} $$

where the term x is the external input term which is only used if the post neuron is an input neuron. The chain thus processes an arbitrary input vector \(\boldsymbol {x} \in R^{n_{I}}\phantom {\dot {i}\!}\) into an output vector in \(\Re ^{n_{O}}\) and this is a highly nonlinear calculation. In Figure 9, we see a chain of six neurons. We see two neurons that accept external input (neurons 0 and 1) and two neurons having external taps (neurons 4 and 5). The set of postsynaptic neurons for a given neuron i is denoted by the symbols \({\mathcal F}(i)\); the letter denotes the feedforward links, of course. Note the self and feedback connection in the set \({\mathcal F}(2) = \{ 2, 5 \}\). Also each neuron can be viewed as a postsynaptic neuron with a set of presynaptic neurons feeding into it: thus, each neuron i has associated with it a set of backward links which will be denoted by \({\mathcal B}(i)\) such as \(\mathcal {B}(5) = \{ 0, 2, 3, 4\}\). In general, the backward link sets can be much richer in connections than this simple example indicates. This model thus uses the computational scheme

$${\fontsize{8.5}{6}\begin{aligned} Y^{post} = \sigma \left(y^{post},o,g \right) \: = \: \sigma \left(x + \sum\limits_{pre \: \in \: {\mathcal B}(post)} \: W_{pre \rightarrow post} \: Y^{pre} \right) \end{aligned}} $$

which can be rewritten \(a^{post} = S \left (x + \sum _{pre \in {\mathcal B}(post)} \: a^{pre} \bullet d^{post}, \: p \right)\) where a indicates the value of a given axon, d, the value of a given dendrite and S, the soma computational engine, which depends on a vector of parameters, p. The term a ^{p r e} ∙d ^{p o s t} denotes the interaction or processing between the post neuron axon and the pre neuron dendrite. Note that the ∙ operation could correspond to a simple multiplication of the pre–axon value d and the post–dendrite value d.

Each of our directed graphs also has node and edge functions associated with it and these functions are time dependent as what they do depends on first and second messenger triggers, the hardware structure of the output neuron and so forth. We therefore could model neural circuitry using a directed graph architecture consisting of computational nodes N and edge functions E which mediate the transfer of information between two nodes. Hence, if N _i and N _j are two computational nodes, then E _i→j would be the corresponding edge function that handles information transfer from node N _i and node N _j . For our purposes, we will assume here that the neural circuitry architecture we describe is fixed, although dynamic architectures can be handled as sequence of directed graphs. We then organize the directed graph using interactions between neural modules (visual cortex, thalamus etc) which are themselves subgraphs of the entire circuit. Once we have chosen a graph to represent the neural circuitry, note the addition of a new neural module is easily handled by adding it and its connections to other modules as a subgraph addition. Hence, at a given level of complexity, if we have the graph \(\boldsymbol {\mathcal {G}(N,E)}\) that encodes the connectivity we wish to model, then the addition of a new module or modules simply generates a new graph \(\boldsymbol {\mathcal {G^{\prime }}(N^{\prime },E^{\prime })}\) for which there are straightforward equations for explaining how G ^′ relates to G which are easy to implement.

The update equations for a given node were given earlier, but now let’s look at them deeper. For the node N _i , let y _i and Y _i denote the input and output from the node, respectively. Then the update strategy with the iteration counter or time labeled at t is given by

$$\begin{aligned} &\boldsymbol{y_{i}}(t+1) = \boldsymbol{I_{i}} + \sum_{j \in \boldsymbol{\mathcal B}(i)} \: \boldsymbol{E_{j \rightarrow i}}(t) \: \boldsymbol{Y_{j}}(t) \quad \text{and} \\ &\boldsymbol{Y_{i}}(t+1) = \boldsymbol{\sigma_{i}}(t)(\boldsymbol{y_{i}}(t)). \end{aligned} $$

where I _i is a possible external input, \(\boldsymbol {\mathcal B}(i)\) is the list of nodes which connect to the input side of node N _i and σ _i (t) is the function which processes the inputs to the node into outputs. This processing function is mutable over time t because second messenger systems are altering how information is processing each time tick. Hence, our model consists of a graph which captures the connectivity or topology of the brain model on top of which is laid the instructions for information processing via the time dependent node and edge processing functions. A simple look at edge processing shows the nodal output which is perhaps an action potential which is transferred without change to a synaptic connection where it initiates a spike in C a ⁺² ions which results in neurotransmitter release. A very simple version of this is to simply assign the value of the edge processing function E _i→j to be the weight W _i,j as is standard in a simple connectionist architecture. Of course, it is more complicated as our graphs allow feedback easily by simply defining the appropriate edge connections. Another approach is to handle feedback terms using a lag ((Werbos 1987) and also see the chapter on lagged chain feed forward architectures (LCFFN) in (Peterson J, BioInformation Processing: A Primer on Computational Cognitive Science in the Springer Cognitive Science and Technology Series, in press). The lagged approach rewrites the feedback structure as a feedforward structure from time t to t+1. The lagged architectures can model much more complicated things than simple feedbacks also, but that is not needed here. Note that this is of theoretical interest as it allows us to repackage any graph model of a neural system with feedback into a new version which is feedforward. Now, so far we have only discussed evaluation of the graph architecture. Usually, there are inputs which must be mapped to specified outputs (a classical training problem) or based on inputs, the edge functions between nodes are used to develop the input to outputs desired (this is a Hebbian approach). We will not discuss this here except to say that the rewriting of the graph into an equivalent feedforward one is useful for the development of computational strategies to hot start a graph model to map given inputs to outputs. The alteration of the parameters in the graph via Hebbian approaches takes a long time and so we can jump to a better parameter start using tools from essentially A x=b techniques, (Peterson 1991), although that is another story too. One can also develop techniques to train a graph and even use subgraphs to do the training and an attempt at this can be found in (Peterson 2014b).