Cable Energy Function of Cortical Axons: Equivalent Formulas

Cortical neurons generally have rich morphologies in dendrite arbor and axonal branches, which make it difficulty in estimate energy consumption during action potential (AP) propagation in neuronal communication. It is an unsolved issue in driving general analytical equations to estimate energy cost for those axons and dendrites with different terminations. Most previous energy calculations of AP-related metabolic cost are still based on the Na + -counting method. Here, we apply principles of physics and mathematical analysis to construct several forms of cable energy function of AP conduction along axons with different boundary conditions. These derived energy equations extend Hodgkin-Huxley theory and prove to be highly more accurate in estimation the energy consumption during AP propagation along cortical axons and dendrites with any kind of ion channels than that using the Na + -counting method. Summary Accurate energy estimation of action potential conduction along axons with different complex terminal conditions is an unsolved issue. We have applied principles of physics and mathematical analysis to derive several forms of cable energy function of action potential conduction along cortical axons with different boundary conditions, and we have proved that these functions are equivalent. The energy calculations of action potential metabolic cost by using our cable energy function is proved to be highly accurate than that based on the Na + -counting method. This mathematical framework allows us to estimate the energy used by AP propagation along cortical axons and dendrites with any kind of ion channels more accurately than that using the Na + -counting method. Accurate calculation of energy consumption of AP conduction may be crucial in the estimation of energy expenditure, from subcellular to whole-brain level. In addition, the analytical formula of energy calculation is valuable in investigating the key factors that influence energy consumption and reveal trade-offs between energetic constraints and neural coding efficiency for individual neurons with rich morphology structures.

In previous studies, estimating the metabolic cost of APs was usually done by recording sodium currents underlying action potentials and calculating the total amount of sodium ions for each AP (Attwell and Laughlin, 2001;Laughlin, 2001;Lennie, 2003;Carter and Bean, 2009;Sengupta et al., 2010;Hallermann et al., 2012;Harris and Attwell, 2012). The total provides an estimate of Na + the number of pump cycles or ATP molecules that the active Na + /K + pump needs to re-establish the resting state of the neuron. However, the -counting method is far from accurate because Na + it does not include all of the factors consuming ATP, thus seriously underestimating the metabolic costs of AP generation (Ju et al., 2016).
To address these shortcoming, energy estimation based on the electrochemical energy function was first performed using a single-compartment Hodgkin-Huxley neuron model (Moujahid et al., 2011;Moujahid and d'Anjou, 2012;Moujahid et al., 2014). Ju et al. developed an energy function for cortical axons (Ju et al., 2016). The analytical approach of the axonal energy function provided an inhomogeneous distribution of metabolic cost along an axon with either uniformly or nonuniformly distributed ion channels. Their results showed that the Na+-counting method severely underestimates energy cost in the cable model by 20-70%. However, this previous analytical work did not satisfactorily consider boundary conditions , as real neuronal axons and dendrites generally have branch point or sealed ends. In this paper, we comprehensively investigate how to calculate the electrochemical energy for a cable equation modeling the cortical axon and derive several equivalent energy functions associated with AP propagation under some assumptions of axon terminations. Then, we compared these results with previous works.

Results
The Cable Energy Function for a Cortical Axon. To precisely estimate the total amount of energy associated with AP generation and propagation along neuronal axons, we refer to the welldeveloped cable theory (Rall, 1989) that has been shown to be consistent with experimental observations (Shu et al., 2006;McCormick et al., 2007;Shu et al., 2007;Yu et al., 2008). and integrating from to for the axon from to end , the energy consumption ( , ) 0 0 of the whole axonal electronic circuit is derived as the following where is the membrane capacitance, is the axonal radius, is the axial intracellular resistivity in Ohms centimeters, and is the AP-stimulating current in micro-amp per stim ( , ) centimeter.
Based on the cable equation that describes how ion currents flow along the cable as well as analysis of the electrical energy in the equivalent circuit, we derive the first energy function of the total electrochemical energy cost of AP propagation in the given cortical axon as the first term of which represents the electrical energy given to the axon at some site, and the other four terms of which represent the total electrical energy accumulated in the equivalent circuit at a time interval . From Eq.
[2], we obtain (0, ) Notably, the right-hand side of Eq. [3] is the same energy function as that derived in our previous work (Ju et al., 2016). Here, we do not add any assumption on the axon terminations; therefore, the energy functions Eq. [2] and Eq. [4] are both applicable to any type of cortical cable model with different terminations (see Methods).
In previous work (Ju et al., 2016), the authors derived Eq. [4] based on the method used in previous references (Moujahid et al., 2011), the viewpoints of which are totally different from ours here.

The Cable Energy Function for a Cortical Axon with Different
Terminations. Next, we applied the above energy function to study the energy function for a cortical cable with killed or sealed terminations. Integrating by parts, we know that the last term of Eq. [3] is For the ends of the cortical axon that are sealed or killed, by the given boundary conditions, we From the mathematical calculation, we know that energy functions Eq. [2], Eq.
[7] are analytically equivalent in this case of boundary conditions, which is also proven by the numerical results (see Table [T2] and Fig. [fig:1]D). Furthermore, when we compared the last term of Eq.
[7], we have in the case of the cortical axon with terminated or sealed ends. For a given site , the ^∈ (0, ) inequality Eq.
[8] implies that the solution , ( is obviously an even number), to ^= 1,2,⋯, are inflection points, and . Immediately, we know that the firing rate at is . Moreover, from the property of the inflection points, we know that the curve of action =^2 potential between and , , is convex, and the other part of the curve is (^, ) concave.
If the end of the axon connects to the soma (lumped-soma termination, see Methods) and = 0 the other end is killed or sealed, that is, or , and ( , ) = 0 As above, we see that the energy functions Eqs. Therefore, the energy cost of AP propagation is obviously less than the cost in other cases.

Discussion
Based on mathematical analysis and physical intuition, we systematically establish the energyfunction method for the cable model of cortical axons. In this paper, we focus on the analytical approach, which is more mathematically accurate. By the analytical equivalence of Eq.
[4], we can obtain identical numerical results as previous work (Ju et al., 2016) showing the following: 1. AP propagation requires more energy than the amount estimated by the point model, and 2. the energy consumption rate of the entire branched axon scales at power of axonal 3 4 volume. See Results for details.
The cable energy functions derived in this paper provide an accurate and applicable way to estimate the energy consumption of electric signals conducted in any type of cortical axon with different terminations. This mathematical framework allows us to estimate the energy used by AP propagation along cortical axons and dendrites with any kind of ion channels more accurately than that using the -counting method. Accurate calculation of energy consumption of AP Na + conduction may be crucial in the estimation of energy expenditure, from subcellular to wholebrain level. In addition, this analytical method of energy calculation can be used to investigate factors that influence energy consumption and reveal trade-offs between energetic constraints and neural coding efficiency for individual neurons with rich morphology structures.

Cable Model of the Cortical Axon
To describe AP propagation along an axon, the cable equation that describes the flow of ion currents along the axon needs to be derived (Gabbiani and Cox, 2010). Figure [fig:1] A and B give the equivalent circuit of the cable model. During AP propagation, the membrane potential, , changes along the -axis, and a longitudinal current, , passing through causes a ( , ) In the context of a realistic modeling of cortical neurons, we can add ion channels to the circuit diagram. Then, in the equation [M3], the term should be replaced by ( , ) = Na + K + L = Na where , , and are the maximal sodium, maximal potassium, and leak conductance per , .
Here, regulates the temperature dependence. Thus, we obtain the cortical cable = That is, the initial data . We input the following form of stimulus to the cable.
where is the Dirac's function, and ( ) ∈ (0, ) where is larger than the threshold. Therefore, we input the initial threshold at the interval soma as an equipotential surface, where is the somatic membrane area, and it is regarded as a single resistance, , and capacitance, , attached to a nerve cylinder as in the case of a natural termination. Then, the boundary condition for a lumped soma at is (see Fig. [fig:1 and if the end at is sealed, = ∂ ∂ ( , ) = 0.
All the parameters (see Table [T1]) in the paper are within the physiological regime (Koch and HC/Biologie, 1999).
For the sake of the readers, we provide the deduction from Eq.

ADDITIONAL INFORMATION
The authors declare no competing financial interests.