footnote

# The Heat of Nervous Conduction$:$ A Thermodynamic Framework

Aymar C. L. de Lichtervelde1,† J. Pedro de Souza2,† Martin Z. Bazant2,3,∗
${}^{1}Department\>of\>Physical\>Chemistry\>\&\>SoftMatter,\break
Wageningen\>University,\>6708\>WG\>Wageningen,\>The\>Netherlands$
${}^{2}\>Department\>of\>Chemical\>Engineering\>and\>^{3}Department\>of\>Mathematics,\break
Massachusetts\>Institute\>of\>Technology,Cambridge,\>MA\>02139,\>USA$

###### Abstract

Early recordings of nervous conduction revealed a notable thermal signature
associated with the electrical signal. The observed production and subsequent
absorption of heat arise from physicochemical processes that occur at the cell
membrane level during the conduction of the action potential. In particular,
the reversible release of electrical energy stored as a difference of
potential across the cell membrane appears as a simple yet consistent
explanation for the heat production, as proposed in the “Condenser Theory.”
However, the Condenser Theory has not been analyzed beyond the analogy between
the cell membrane and a parallel-plate capacitor, i.e. a condenser, which
cannot account for the magnitude of the heat signature. In this work, we use a
detailed electrostatic model of the cell membrane to revisit the Condenser
Theory. We derive expressions for free energy and entropy changes associated
with the depolarization of the membrane by the action potential, which give a
direct measure of the heat produced and absorbed by neurons. We show how the
density of surface charges on both sides of the membrane impacts the energy
changes. Finally, considering a typical action potential, we show that if the
membrane holds a bias of surface charges, such that the internal side of the
membrane is 0.05 C m-2 more negative than the external side, the size of the
heat predicted by the model reaches the range of experimental values. Based on
our study, we identify the change in electrical energy of the membrane as the
primary mechanism of heat production and absorption by neurons during nervous
conduction.

##  I Introduction

###  I.1 Thermodynamics of nervous conduction

Besides the electrical responses classically measured in electrophysiological
experiments Hodgkin  _et al._ (1952); Hodgkin and Huxley (1952); Neher and
Sakmann (1992); Hamill  _et al._ (1981), the action potential is accompanied
with a production and a subsequent absorption of heat Howarth  _et al._
(1979), changes in optical properties Tasaki  _et al._ (1968), and mechanical
deformations Tasaki  _et al._ (1989); Gonzalez-Perez  _et al._ (2016). These
thermal, optical, and mechanical responses are macroscopic signatures of the
physicochemical processes occurring at the cell membrane level during the
action potential, such as the transport of ions through ion channels in the
membrane Hille (2001) or the elastic deformation of the membrane El Hady and
Machta (2015). While such physicochemical signatures are associated with the
electrical signal, classical electrical circuit models cannot accurately
capture them because the circuit models neglect the microscopic physics at the
membrane level.

Here we examine the thermal response of nervous conduction, by resolving the
microscopic physics of the membrane and its surrounding electrical double
layers. We start by reviewing experimental and theoretical backgrounds on the
heat signature of nervous conduction. Then we apply an equilibrium-
thermodynamics framework to calculate the electrical energy that is stored by
the membrane and the surrounding double layers and released into heat during
the passage of the action potential. Finally, based on typical
neurophysiological parameters, we show that the reversible release of
electrical energy offers a plausible explanation for the heat of nervous
conduction.

###  I.2 Heat production and absorption by neurons: the experimental context

A substantial experimental record shows that the propagation of the action
potential along neurons is accompanied by the release of a small amount of
heat, immediately followed by the absorption of a similar amount of heat by
the neurons Abbott  _et al._ (1958); Howarth and Ritchie (1968); Ritchie and
Keynes (1985); Howarth  _et al._ (1979). Successfully measured for the first
time in 1925 Ritchie (1973), the heat of nervous conduction has been most
extensively investigated between the 1950s and 1980s by contemporaries and
colleagues of Hodgkin and Huxley Abbott  _et al._ (1958); Howarth and Ritchie
(1968); Ritchie and Keynes (1985); Howarth  _et al._ (1979), the pioneers of
modern neurophysiology. All neurons possess a similar excitable membrane, and
heat production is likely a universal feature of nervous conduction. However,
the thermal signals are most easily measured in nerve fibers that have a high
surface-to-volume ratio. The thermal signals are indeed extremely small, and
it appears that the heat flux is proportional to the axon membrane area
Howarth  _et al._ (1979). The garfish olfactory nerve, for example, is an
excellent candidate for recording the heat of nervous conduction: it is made
of several millions of fibers of 0.25 $\text{\,}\mathrm{\SIUnitSymbolMicro m}$
in diameter, totals a membrane area of 6.5 m2 per g of nerve Easton (1971),
and releases heat on the order of 1 mJ g-1 Howarth  _et al._ (1979). When
expressed per total membrane area, the size of the heat remains on the same
order of magnitude from one organism to the other (60 - 180
$\text{\,}\mathrm{\SIUnitSymbolMicro J}$ m-2 Howarth  _et al._ (1979)).

To understand the origin of the thermal signals, scientists attempted to
correlate them with the electrical signals Abbott  _et al._ (1958); Howarth
and Ritchie (1968); Howarth  _et al._ (1979). Notably, Howarth  _et al._
(1979) successfully reconstructed the true temperature change that occurs in
neurons from recorded heat responses, and showed that the time course of the
temperature changes closely matches the one of the membrane potential during
the action potential. Such finding gave support to the “Condenser Theory.”

###  I.3 The Condenser Theory

The Condenser Theory offers a simple explanation for the heat production and
absorption: it attributes them to the reversible release of electrical energy
stored across the cell membrane Abbott  _et al._ (1958); Howarth and Ritchie
(1968); Howarth  _et al._ (1979); Ritchie and Keynes (1985). At rest, the
membrane of neurons holds a difference of electric potential, called “membrane
potential.” An action potential occurs when the membrane potential at a
specific location rapidly rises (depolarization) and falls (repolarization),
due to the opening of ion channels Hille (2001). The Condenser Theory states
that as the action potential depolarizes the membrane, the electrical energy
stored across the cell membrane is released into heat. Conversely, upon
repolarization of the membrane to its resting potential, the membrane’s
electrical energy is restored at the expense of some of the thermal energy of
the ions in the surrounding solutions, which accounts for the heat absorption
phase, in symmetry with the production phase. The membrane is seen as a
capacitor (or a "condenser"), hence this explanation for the heat of nervous
conduction is known as the Condenser Theory. In the first developments of the
Condenser Theory, the amount of heat reversibly exchanged between the membrane
and its surroundings was calculated as the free energy of a parallel-plate
capacitor Howarth  _et al._ (1979); Ritchie and Keynes (1985):

$\Delta{F}=\frac{1}{2}c_{\mathrm{m}}({V_{\mathrm{m}}}^{2}-{V_{\mathrm{m},0}}^{2})$
(1)

where $\Delta{F}$ is the free energy change (J m-2), $c_{\mathrm{m}}$ (F m-2)
the membrane capacitance, $V_{\mathrm{m}}$ the membrane potential (the inside
potential minus the outside potential, cf. Fig. 1) and $V_{\mathrm{m},0}$ the
resting potential (typically $-70$ mV Weiss (1996)).

###  I.4 Arguments to explain the missing heat

Though in qualitative agreement with the experimental records, Eq. (1) only
predicts a quarter of the heat that is measured. Realising this, Howarth  _et
al._ (1979) and Ritchie and Keynes (1985) suggested that the free energy
should be calculated based on the local value of potential falling on each
side of the membrane (cf. $\phi_{\mathrm{t}}$ in Fig. 1), rather than on the
potential values in the internal and external bulk solutions (cf.
$V_{\mathrm{m}}$ in Fig. 1):

$\Delta{F}_{Ritchie}=\frac{1}{2}\>c_{\mathrm{m}}({\phi_{\mathrm{t}}}^{2}-{\phi_{\mathrm{t},0}}^{2})$
(2)

In particular, it was pointed out that the presence of an uneven distribution
of negative surface charges on the membrane (more charges on the internal side
than on the external side) would increase the transmembrane potential
difference and lead to more heat being evolved Howarth  _et al._ (1979);
Ritchie and Keynes (1985). Unfortunately, Howarth  _et al._ (1979) and Ritchie
and Keynes (1985) did not provide a careful derivation for Eq. (2), nor did
they explore the physics of the cell membrane and its surface charges beyond
the analogy with a parallel-plate capacitor.

A second argument invoked by several authors to bridge the gap between
predicted and measured heats concerns entropy changes presumed to occur inside
the lipid bilayer (i.e. the membrane) when the electric field across the
membrane relaxes Howarth and Ritchie (1968); Ritchie (1973); Howarth  _et al._
(1979); Ritchie and Keynes (1985). Specifically, they proposed that the total
energy change associated with the depolarization of the membrane could differ
significantly from the free energy change $\Delta{F}$, by an entropy
contribution $T\Delta{S}$. These changes were calculated proportionally to the
free energy term, based on the temperature dependence of the membrane
capacitance:

$T\Delta{S}=\Delta{F}\>\frac{T}{c_{\mathrm{m}}}\>\frac{\partial{c_{\mathrm{m}}}}{\partial{T}}$
(3)

where $\Delta{S}$ is the entropy change and $T$ the temperature. Eq. (3)
predicts an additional heating of the lipid membrane if $T\Delta{S}$ and
$\Delta{F}$ have the same sign, otherwise it predicts a cooling. According to
Ritchie and Keynes (1985) and Ref. Taylor  _et al._ (1962), the value of
${T/c_{\mathrm{m}}}\>{\partial{c_{\mathrm{m}}}}/{\partial{T}}$ is positive,
between 2 and 4, which would bring a total warming that is 3 to 5 times higher
than if the heat was derived only from the release of free energy stored in
the membrane capacity. However, no rigorous derivation for Eq. 3 could be
found in literature. Furthermore, the prediction of Ritchie and Keynes (1985)
seems difficult to reconcile with recent measurements of the temperature
dependence of the dielectric permittivity
(${\partial{\varepsilon}}/{\partial{T}}$) of fatty acids, the carbon chains
that form the cell membrane. Indeed, ${\partial{\varepsilon}}/{\partial{T}}$
appears to be negative Plaksin  _et al._ (2018); Lizhi  _et al._ (2008), and
if we calculate the membrane capacitance $c_{\mathrm{m}}$ as proportional to
its dielectric permittivity $\varepsilon_{\mathrm{m}}$ we expect
${\partial{c_{\mathrm{m}}}}/{\partial{T}}$ to be negative rather than
positive, which contradicts the argument of Ritchie and Keynes (1985).

More than a simple parallel-plate capacitor, the axon membrane consists of a
lipid bilayer with surface charges and electrical double layers forming on
each side Genet  _et al._ (2000). The analogy with a condenser offers a too
limited description of the membrane to verify the correctness of the arguments
reviewed above. It does not allow to judge which of Eqs. (1) or (2) describe
correctly the free energy change during the depolarization of the membrane. In
addition, as shown above, how entropy changes inside the membrane could
contribute to the heat production still needs to be understood.

To assess whether electrical energy changes constitute a plausible explanation
for the heat production and absorption by neurons, we will now derive the
changes in electric free energy and entropy that accompany the action
potential, based on a detailed electrostatic model of the membrane, its
surface charges, and double layers.

##  II Theory

###  II.1 Electrostatic model of the charged lipid bilayer

We use the coupled electrostatic model proposed by Genet  _et al._ (2000),
which applies Poisson-Boltzmann theory on either side of the cell membrane.
Fig. 1 shows the qualitative electric potential profile surrounding and inside
a cell membrane that holds surface charges on the internal and external sides
($\sigma_{\mathrm{i}}$ and $\sigma_{\mathrm{o}}$, respectively). By
convention, we use the symbols $-\infty$ and $+\infty$ to denote the
(arbitrary) limits between the diffuse layer and bulk regions, in the internal
and external solutions, respectively. Note that in physiological conditions,
diffuse layers extend over a few nano-meters at most (the Debye length is 0.6
nm). The membrane core is located between $x=-\delta$ and $x=0$, where
$\delta$ is the thickness of the membrane. The $\phi_{\mathrm{i}}$,
$\phi_{\mathrm{m}}$ and $\phi_{\mathrm{o}}$ variables represent the potential
in the internal solution, membrane core and external solution, respectively.
The membrane potential is defined as the difference between the potential in
the internal and external bulk solutions,
$V_{\mathrm{m}}=\phi(-\infty)-\phi(+\infty)$, whereas the transmembrane
potential $\phi_{\mathrm{t}}$ is the potential difference between the internal
and external surfaces of the membrane,
$\phi_{\mathrm{t}}=\phi_{\mathrm{i}}(-\delta)-\phi_{\mathrm{o}}(0)$. Note that
due to the presence of surface charges on each side of the membrane, the local
difference of potential $\phi_{\mathrm{t}}$ can differ significantly from the
membrane potential $V_{\mathrm{m}}$.

Figure 1: Profile of the electric potential across the cell membrane and the
diffuse layers on each side. The minus signs in white denote surface charges
($\sigma_{\mathrm{i}}$ inside and $\sigma_{\mathrm{o}}$ outside), while the
charges in solution are drawn in blue and red. The dotted lines delimit
thermodynamic domains (see Sec. II.2). At rest, the membrane holds a potential
difference ($V_{\mathrm{m}}=-70$ mV). Upon excitation, cations cross the
membrane through ion channels (not depicted here) and $V_{\mathrm{m}}$ jumps
to positive values. Simultaneously, electrical energy is dissipated as heat
($Q<0$).

We assume that equilibration of diffuse layers with the bulk electrolytes is
fast compared to the dynamics of the action potential (this is verified in
Section VII.1), which allows us to describe the concentration of ions in the
diffuse layers close to the membrane with the Boltzmann distribution

$\displaystyle c_{\mathrm{i},j}(x)$
$\displaystyle=c_{\mathrm{i},j}(b)\>\exp{\\{-\frac{z_{j}\>F\>\phi_{\mathrm{i}}}{R\>T}\\}},$
(4) $\displaystyle c_{\mathrm{o},j}(x)$
$\displaystyle=c_{\mathrm{o},j}(b)\>\exp{\\{-\frac{z_{j}\>F\>(\phi_{\mathrm{o}}-V_{\mathrm{m}})}{R\>T}\\}},$
(5)

where $c_{\mathrm{i},j}$ and $c_{\mathrm{o},j}$ are the concentrations of
species $j$ inside and outside, respectively, and $RT/F$ is the thermal
voltage ($RT/F=23.5$ mV at $T=273$ K). Applying Poisson’s equation to each
compartment gives

$\displaystyle\frac{d^{2}{\phi_{\mathrm{i}}}}{dx^{2}}$
$\displaystyle=-\frac{\rho_{\mathrm{i}}}{\varepsilon},$ (6)
$\displaystyle\frac{d^{2}{\phi_{\mathrm{m}}}}{dx^{2}}$ $\displaystyle=0,$ (7)
$\displaystyle\frac{d^{2}{\phi_{\mathrm{o}}}}{dx^{2}}$
$\displaystyle=-\frac{\rho_{\mathrm{o}}}{\varepsilon},$ (8)

where $\rho$ is the density of free charges (C m-3), and $\varepsilon$ the
dielectric permittivity of the internal and external solutions (water). Note
that we assume that the free charge density is zero inside the membrane
($\rho_{\mathrm{m}}=0$), with a zero ion concentration inside the membrane. As
$\rho_{\mathrm{i}}=\sum{z_{j}\>F\>c_{\mathrm{i},j}}$ and
$\rho_{\mathrm{o}}=\sum{z_{j}\>F\>c_{\mathrm{o},j}}$, we obtain the following
Poisson-Boltzmann equations:

$\displaystyle\frac{d^{2}\phi_{\mathrm{i}}}{dx^{2}}(x)$
$\displaystyle=-\frac{F}{\varepsilon}\sum^{N}_{j}\>z_{j}\>c_{\mathrm{i},j}(b)\>\exp{\\{-\frac{z_{j}F\>\phi_{\mathrm{i}}(x)}{R\>T}\\}},$
(9) $\displaystyle\frac{d^{2}\phi_{\mathrm{o}}}{dx^{2}}(x)$
$\displaystyle=-\frac{F}{\varepsilon}\sum^{N}_{j}\>z_{j}\>c_{\mathrm{o},j}(b)\>\exp{\\{-\frac{z_{j}F\>[\phi_{\mathrm{o}}(x)-V_{\mathrm{m}}]}{R\>T}\\}},$
(10)

which we solve numerically, using boundary conditions given by Maxwell’s
equations at an interface (Eqs. (15) and (16)). We now report the
electrostatic relations necessary to derive energy changes in following
sections. At rest, the membrane acts as a dielectric medium (ion channels are
closed), storing a capacitive charge that we define as $-q$ in the internal
solution and $+q$ in the external solution. In each compartment, this
capacitive charge can be expressed as the sum of the charge that counter-
balances the charges that belong to the surface of the membrane (that is
$-\sigma_{\mathrm{i}}$ inside and $-\sigma_{\mathrm{o}}$ outside, both with
units C m-2) and the total mobile charge in solution, which we obtain by
integrating the free charge density over the diffuse layers:

$\displaystyle-q$
$\displaystyle=\sigma_{\mathrm{i}}+\int_{-\infty}^{-\delta}{\rho_{\mathrm{i}}}\>dx,$
(11) $\displaystyle q$
$\displaystyle=\sigma_{\mathrm{o}}+\int_{0}^{\infty}{\rho_{\mathrm{o}}}\>dx.$
(12)

Applying Poisson’s equation to the integrals above, we can relate the slope of
the membrane potential at each interface to the total charge density in the
internal and external solutions

$\displaystyle\frac{d\phi_{\mathrm{i}}}{dx}(-\delta)$
$\displaystyle=\frac{1}{\varepsilon}(\sigma_{\mathrm{i}}+q),$ (13)
$\displaystyle\frac{d\phi_{\mathrm{o}}}{dx}(0)$
$\displaystyle=-\frac{1}{\varepsilon}\>\Big{(}\sigma_{\mathrm{o}}-q\Big{)},$
(14)

Boundary conditions for electric potential at the two membrane-solution
interfaces are given by

$\displaystyle\varepsilon\frac{d\phi_{\mathrm{i}}}{dx}(-\delta)-\varepsilon_{\mathrm{m}}\frac{d\phi_{\mathrm{m}}}{dx}(-\delta)$
$\displaystyle=\sigma_{\mathrm{i}},$ (15)
$\displaystyle\varepsilon_{\mathrm{m}}\frac{d\phi_{\mathrm{m}}}{dx}(0)-\varepsilon\frac{d\phi_{\mathrm{o}}}{dx}(0)$
$\displaystyle=\sigma_{\mathrm{o}},$ (16)

where $\varepsilon_{\mathrm{m}}$ is the dielectric permittivity of the
membrane (F m-1). By substituting Eq. (13) into (15), and Eq. (14) into (16)
we can deduce that

$\frac{q}{\varepsilon_{\mathrm{m}}}=\frac{d\phi_{\mathrm{m}}}{dx}(-\delta)=\frac{d\phi_{\mathrm{m}}}{dx}(0).$
(17)

As the electric field is assumed to be constant inside the membrane, Eq. (17)
gives

$\frac{q}{\varepsilon_{\mathrm{m}}}=\frac{\phi_{\mathrm{m}}(0)-\phi_{\mathrm{m}}(-\delta)}{\delta}\overset{\Delta}{=}-\frac{\phi_{\mathrm{t}}}{\delta}.$
(18)

We finally obtain an expression that relates the capacitive charge to the
membrane’s capacitance ($c_{\mathrm{m}}$, in F m-2) and the transmembrane
potential:

$q=-\frac{\varepsilon_{\mathrm{m}}}{\delta}\>\phi_{\mathrm{t}}=-c_{\mathrm{m}}\>\phi_{\mathrm{t}},$
(19)

with $c_{\mathrm{m}}\overset{\Delta}{=}{\varepsilon_{\mathrm{m}}}/{\delta}$,
identical to the result from Genet  _et al._ (2000).

###  II.2 Thermodynamic definitions

To relate electrical energy changes at the membrane level to the exchange of
heat with the surroundings, we need to define the membrane as a thermodynamic
system. We divide the space into three domains (see Fig. 1):
$\Omega_{\mathrm{M}}$ is the “membrane domain”, i.e. the region comprised of
the membrane and the diffuse layers that form in the internal and external
solutions, while $\Omega_{\mathrm{b,i}}$ and $\Omega_{\mathrm{b,o}}$ denote
the internal and external bulk solutions, respectively. The first law of
thermodynamics, $\Delta{U_{\mathrm{el}}}=Q+W_{\mathrm{el}}$, applies, in which
$\Delta{U_{\mathrm{el}}}$ is the internal energy change associated with the
variation of the membrane potential from $V_{\mathrm{m},0}$ to
$V_{\mathrm{m}}$, $Q$ is the heat added to $\Omega_{\mathrm{M}}$ ($Q<0$ when
heat is dissipated) and $W_{\mathrm{el}}$ is the electrical work done on
$\Omega_{\mathrm{M}}$ by the surroundings. Since the capacitive charges $+q$
and $-q$ are confined to the diffuse layers close to the membrane (the bulk
solutions are electroneutral), no electrical work is done on
$\Omega_{\mathrm{M}}$ by the surrounding domains $\Omega_{\mathrm{b,i}}$ and
$\Omega_{\mathrm{b,o}}$. The first law thus becomes

$\Delta{U_{\mathrm{el}}}=Q.$ (20)

The internal energy of the membrane system can be expressed as a sum of free
energy and entropy-related energy:

$\Delta{U_{\mathrm{el}}}=\Delta{F_{\mathrm{el}}}+T\Delta{S_{\mathrm{el}}},$
(21)

where $\Delta{F_{\mathrm{el}}}$ and $\Delta{S_{\mathrm{el}}}$ are respectively
the free energy and entropy change with respect to the resting state
($V_{\mathrm{m}}=-70$ mV). Thus, based on the aforementioned definitions, heat
will be released ($Q<0$) from the membrane domain $\Omega_{\mathrm{M}}$ to the
surroundings, as the (electric) internal energy of the membrane domain
decreases ($\Delta{U_{\mathrm{el}}}<0$). In the following sections, we will
derive the free energy and entropy changes in the membrane domain as a
function of the membrane potentials $V_{\mathrm{m}}$ and $\phi_{\mathrm{t}}$.
The sum of these energies will give the internal energy change (by Eq. (21))
associated with the (de)polarization of the membrane and therefore the
quantity of heat that is reversibly released from the membrane domain
$\Omega_{\mathrm{M}}$.

###  II.3 Electric free energy

The electric free energy of a linear dielectric medium is equal to its field
energy Overbeek (1990); Jackson (1999); Landau  _et al._ (2013), that is, in
one dimension

$F_{{\mathrm{el}}}=\frac{1}{2}\int^{+\infty}_{-\infty}{E.D}\>dx,$ (22)

where $F_{{\mathrm{el}}}$ is the electric free energy (J m-2), $E$ the
electric field (V m-1) and $D=\varepsilon\>E$ the displacement field (C m-2).
Interestingly, the field energy can be equated with the amount of heat
dissipated by ionic currents in an electric field, using Maxwell’s equation
for Ampere’s law (this is shown in Section VII.3). Note that the field energy
is often regarded as an internal energy ($U_{\mathrm{el}}$), however, this is
only true in a primitive model that considers the medium as structureless
Frohlich (1968); Overbeek (1990). In a more refined model that takes into the
effect of the electric field on the entropy of the dielectric medium (see
SectionII.4), the field energy must be regarded as a free energy
($F_{\mathrm{el}}$) Frohlich (1968); Overbeek (1990). Applying Eq. (22) to the
$\Omega_{\mathrm{M}}$ domain, we obtain the electric free energy as a function
of the membrane potential $V_{\mathrm{m}}$. We obtain a free energy
contribution from the double layers, which is

$\begin{split}{F}^{\mathrm{DL}}_{\mathrm{el}}=&\frac{1}{2}\bigg{(}\int_{-\infty}^{-\delta}{\rho_{\mathrm{i}}(\phi_{\mathrm{i}}-V_{\mathrm{m}})}dx+\sigma_{\mathrm{i}}\>(\phi_{\mathrm{i}}(-\delta)-V_{\mathrm{m}})\\\
&+\int^{+\infty}_{0}{\rho_{\mathrm{o}}\>\phi_{\mathrm{o}}\>}dx+\sigma_{\mathrm{o}}\>\phi_{\mathrm{o}}(0)\bigg{)},\end{split}$
(23)

and one from the membrane capacitance, which is

${F}^{\mathrm{m}}_{\mathrm{el}}=\frac{1}{2}\>c_{\mathrm{m}}\>\phi_{\mathrm{t}}\>V_{\mathrm{m}},$
(24)

with
${F}_{\mathrm{el}}={F}^{\mathrm{DL}}_{\mathrm{el}}+{F}^{\mathrm{m}}_{\mathrm{el}}$.
The first and third terms in Eq. (23) correspond to the energy of bulk charges
in the diffuse layers of internal and external solutions and the second and
fourth to the energy of the surface charges fixed onto the membrane. As shown
by Eq. (24), the free energy contribution of the membrane capacitance is
${1}/{2}\>c_{\mathrm{m}}\>\phi_{\mathrm{t}}\>V_{\mathrm{m}}$, and not
${1}/{2}c_{\mathrm{m}}\>\phi_{\mathrm{t}}^{2}$ as proposed in Refs. Howarth
_et al._ (1979) and Ritchie and Keynes (1985). The latter overestimates free
energy changes, as compared to Eq. (24), see Fig. 6 in Section VII.6. The full
derivation of Eqs. (23) and (24) is presented Section VII.4.

Finally, we write the free energy changes associated with the depolarisation
of the membrane as
$\Delta{F^{\mathrm{DL}}_{\mathrm{el}}}={F^{\mathrm{DL}}_{\mathrm{el}}}-{F^{\mathrm{DL}}_{\mathrm{el,0}}}$
and
$\Delta{F^{\mathrm{m}}_{\mathrm{el}}}={F^{\mathrm{m}}_{\mathrm{el}}}-{F^{\mathrm{m}}_{\mathrm{el,0}}}$,
where subscripts “0” mark the free energies calculated at the resting membrane
potential, $V_{\mathrm{m,0}}=-70$ mV.

###  II.4 Entropy associated with the electric field

We consider two entropy terms: entropy changes associated with the
polarization of water in the diffuse layers and entropy changes in the
membrane.

####  II.4.1 In the diffuse layers

The electric field orders water dipoles in the diffuse layers, which decreases
entropy. The change of entropy associated with the alignment of dipoles in a
dielectric medium (water in our case) is related to the electric free energy
by Frohlich (1968):

$T\Delta{S}^{\mathrm{DL}}_{\mathrm{el}}=\frac{T}{\varepsilon}\frac{\partial{\varepsilon}}{\partial{T}}\Delta{F^{\mathrm{DL}}_{\mathrm{el}}}$
(25)

where $\varepsilon$ is the dielectric permittivity of the medium. The value of
${T}/{\varepsilon}\>{\partial{\varepsilon}}/{\partial{T}}$ for water is -1.17
at 0°C and -1.4 at human body temperature (37°C) Floriano and Nascimento
(2004); Lide (2004).

####  II.4.2 In the membrane

Similarly, entropy changes inside the lipid membrane have been proposed based
on the temperature dependence of the membrane capacitance Howarth and Ritchie
(1968); Howarth  _et al._ (1979); Ritchie and Keynes (1985):

$T\Delta{S^{\mathrm{m}}_{\mathrm{el}}}=\frac{T}{c_{\mathrm{m}}}\>\frac{\partial{c_{\mathrm{m}}}}{\partial{T}}\>\Delta{F^{\mathrm{m}}_{\mathrm{el}}}.$
(26)

To verify that Eq. (26) holds, we adapted the derivation of Eq. (25), in which
entropy is a function of ${\partial{\varepsilon}}/{\partial{T}}$, to the case
in which entropy is a function of ${\partial{c_{\mathrm{m}}}}/{\partial{T}}$,
the temperature dependence of the membrane capacitance. As shown in Section
VII.5, Eq. (26) holds, under the assumption that the membrane capacitance has
a linear dependence on temperature, but is constant with potential.
Interestingly, it appears from recent experiments that the temperature
dependence of the membrane capacitance arises from the variation of the lipid
bilayer’s dimensions (thickness, $\delta$, and area, $A$, per lipid molecule)
with temperature, rather than from the one of its dielectric permittivity
(${\partial{\varepsilon_{\mathrm{m}}}}/{\partial{T}}\simeq 0$) Plaksin  _et
al._ (2018); Norbert Kucerka and Katsaras (2011); Szekely  _et al._ (2011);
Pan  _et al._ (2008):

$\frac{\partial{c_{\mathrm{m}}}}{\partial{T}}=\frac{\partial{(\varepsilon_{\mathrm{m}}\>A/\delta)}}{\partial{T}}\simeq\varepsilon_{\mathrm{m}}\frac{\partial{(A/\delta)}}{\partial{T}}.$
(27)

Based on recent measurements of these dimensional changes Norbert Kucerka and
Katsaras (2011); Szekely  _et al._ (2011); Pan  _et al._ (2008) Plaksin  _et
al._ (2018) have pointed out that the temperature dependence of the membrane
capacitance remains close to $+0.3\%$/°C across several cellular types and
artificial lipid membranes, suggesting that the rate of thermal response of
the membrane is universal.

###  II.5 Parameters

Two important parameters in this model are the surface charge density on the
interior ($\sigma_{\mathrm{i}}$ ) and exterior ($\sigma_{\mathrm{o}}$) sides
of the membrane. Here we report ranges of values found in literature and then
choose baselines for these two parameters. Hille (2001) compiled experimental
data on excitable membranes, showing that the surface charge density
(extracellular and intracellular) varies from $-0.04$ to $-0.16$ C m-2. Other
estimates give a wider range of values, from $-0.002$ to $-0.37$ C m-2
Lakshminarayanaiah and Murayama (1975). In this work, the external surface
charge density will be fixed to $\sigma_{\mathrm{o}}=-0.05$ C m-2, and we will
evaluate three cases for the internal surface charge density:
$\sigma_{\mathrm{i}}=-0.05$, $-0.1$ and $-0.15$ C m-2. There are several lines
of evidence suggesting that neurons have more negative surface charges on the
internal side of the membrane than on the external one. In rat cortical
neurons for instance, Plaksin  _et al._ (2018) estimated this surface charge
bias to be $\sigma_{\mathrm{i}}-\sigma_{\mathrm{o}}=-0.1$ C m-2. Further
support for this hypothesis arises from the uneven distribution of
phospholipids between the two sides of the membrane. In particular,
phosphatidylserine, the most abundant negatively charged phospholipid in cell
membranes, is found exclusively on the internal side of the cell membrane of
neurons, where it has key signaling functions Kim  _et al._ (2014). Other
parameters used in the model are reported in Table 1.

Parameter | Value | Unit | Source
---|---|---|---
$\delta$ | 3 | nm | Genet  _et al._ (2000)
$c_{\mathrm{m}}$ | 9 | mF m-2 | Genet  _et al._ (2000)
$V_{\mathrm{m}}$ | -70 | mV | Weiss (1996); Hille (2001); Genet  _et al._ (2000)
${1}/{\varepsilon}\>{\partial{\varepsilon}}/{\partial{T}}$ | \- 0.43 at 0°C |  $\%$K-1 | Floriano and Nascimento (2004); Lide (2004)
${1}/{c_{\mathrm{m}}}\>{\partial{c_{\mathrm{m}}}}/{\partial{T}}$ | \+ 0.3 |  $\%$K-1 | Plaksin  _et al._ (2018)
$c_{\ce{Na+}}$ | 5 / 145 | mM | Johnston and Wu (1994)
$c_{\ce{K+}}$ | 140 / 5 | mM | Johnston and Wu (1994)
$c_{\ce{Ca^{2}+}}$ | 0.0001 / 2.5 | mM | Johnston and Wu (1994)
$c_{\ce{Cl-}}$ | 145 / 155 | mM | Johnston and Wu (1994)
$T$ | 273 | K | Howarth  _et al._ (1979)
Table 1: Parameters used in the electrostatic model and energy calculations.
Concentrations are given in the format: internal solution / external solution.

##  III Results

###  III.1 Free energy changes

The change in electric free energy with membrane potential is depicted in Fig.
2, in the membrane ($\Delta{F^{\mathrm{m}}_{\mathrm{el}}}$) and in diffuse
layers ($\Delta{F^{\mathrm{DL}}_{\mathrm{el}}}$), with and without surface
charges on the membrane. First, when there are no surface charges, the free
energy follows a parabola centred around $V_{\mathrm{m}}$ = 0 mV. The diffuse
layers bring a relatively negligible contribution to the free energy in this
case. Interestingly, the presence of an equal amount of surface charges on
each side of the membrane ($\sigma_{\mathrm{i}}=\sigma_{\mathrm{o}}=-0.05$ C
m-2) results in almost no alteration of the energy changes as compared to the
zero surface charge case. However, as more surface charges are present on the
internal side than on the external side of the membrane
($\sigma_{\mathrm{i}}=2\>\sigma_{\mathrm{o}}$ and $3\>\sigma_{\mathrm{o}}$ ),
both $\Delta{F^{\mathrm{m}}_{\mathrm{el}}}$ and
$\Delta{F^{\mathrm{DL}}_{\mathrm{el}}}$ follow a steeper decrease, such that
more free energy is released with increasing $V_{\mathrm{m}}$.

Figure 2: Change in electric free energy in the membrane
($\Delta{F}^{\mathrm{m}}_{\mathrm{el}}$) and in diffuse layers
($\Delta{F}^{\mathrm{DL}}_{\mathrm{el}}$) as a function of membrane potential,
for an equal amount of surface charges
($\sigma_{\mathrm{i}}=\sigma_{\mathrm{o}}=-0.05$ C m-2) and with a surface
charge bias ($\sigma_{\mathrm{i}}=2\>\sigma_{\mathrm{o}}$ and
$3\>\sigma_{\mathrm{o}}$). When a surface charge bias is present, more free
energy is released, both in the membrane and diffuse layers.

###  III.2 Entropy changes

Entropy changes, depicted in Fig. 3, are proportional to the free energy
changes presented above (by Eqs. (25) and (26)). As the membrane depolarizes
(increasing $V_{\mathrm{m}}$), the entropy in the membrane
($T\Delta{S}^{\mathrm{m}}_{\mathrm{el}}$) decreases, while the one in diffuse
layers ($T\Delta{S}^{\mathrm{DL}}_{\mathrm{el}}$) increases. Both entropy
changes follow the same trend with surface charge distribution as the free
energy changes in Fig. 2.

Figure 3: Entropy changes in the membrane
($T\Delta{S}^{\mathrm{m}}_{\mathrm{el}}$) and diffuse layers
($T\Delta{S}^{\mathrm{DL}}_{\mathrm{el}}$) with no surface charges, an equal
amount of surface charges ($\sigma_{\mathrm{i}}=\sigma_{\mathrm{o}}=$-0.05 C
m-2) and with a surface charge bias
($\sigma_{\mathrm{i}}=2\>\sigma_{\mathrm{o}}$ and $3\>\sigma_{\mathrm{o}}$).
The presence of a surface charge bias makes the entropy in the membrane
decrease more upon depolarization, whereas it makes the entropy in diffuse
layers increase.

###  III.3 Internal energy change, heat production and temperature change

The internal energy change occurring inside the $\Omega_{\mathrm{M}}$ domain,
i.e the sum of the free and the entropic energy changes, is depicted in Fig. 4
according to the different scenarios of surface charge distribution used until
now. The internal energy change provides a direct measure of the heat produced
and absorbed by the nerve during the course of the action potential (Eq.
(20)): an internal energy decrease ($\Delta{U}<0$) corresponds to a release of
heat by the membrane domain ($Q<0$), whereas an internal energy increase
($\Delta{U}>0$) corresponds to an absorption of heat by the membrane domain
($Q>0$). An idealized action potential (modelled with a normal distribution
function, for simplicity) and the corresponding heat profile are depicted in
Fig. 5. We find that the presence of negative surface charges on the membrane
leads to a more important decrease in internal energy (Fig. 4) and thus heat
production (Fig. 5), especially when the membrane holds more fixed charges on
its inside than outside. Interestingly, the bottom curve in Fig. 4 shows that
when surface charges are distributed unevenly, the internal energy must not
rise immediately after that $V_{\mathrm{m}}$ takes positive values. In other
words, the membrane can release heat even when the membrane potential
overshoots to positive values. This is reflected in Fig. 5 by the progressive
disappearance of the “notch” in the energy profiles as $\sigma_{\mathrm{i}}$
becomes more negative than $\sigma_{\mathrm{o}}$.

Figure 4: Internal energy changes of the membrane system, from top to bottom:
without surface charge ($\sigma_{\mathrm{i}}=\sigma_{\mathrm{o}}=$ 0), with an
equal amount of surface charges
($\sigma_{\mathrm{i}}=\sigma_{\mathrm{o}}=-0.05$ C m-2), with a moderate
surface charge bias ($\sigma_{\mathrm{i}}=2\>\sigma_{\mathrm{o}}$) and a
bigger one ($\sigma_{\mathrm{i}}=3\>\sigma_{\mathrm{o}}$). The internal energy
release increases with the presence of negative surface charges on each side
of the membrane, and significantly more as the surface charge difference
between the internal and external side of the membrane rises.  Figure 5: Left:
an action potential depolarizes the membrane, starting from a resting
potential of $V_{\mathrm{m}}$ = -70 mV at time 0, the membrane depolarizes to
$V_{\mathrm{m}}$ = +20 mV at the peak of the action potential, before
decreasing back to the resting potential. Right: concomitantly with the action
potential, heat ($-Q$) is produced and absorbed, increasing the nerve
temperature ($\Delta{T}$). See Section VII.2 for the calculation of
temperature increase as a function of the heat released.

Finally, based on Fig. 5, we predict that a typical depolarization of the
membrane from -70 mV to +20 mV Weiss (1996); Dudel  _et al._ (2012) leads to a
heat production of $-Q=40$ to $70\text{\,}\mathrm{\SIUnitSymbolMicro J}$ m-2,
depending on the magnitude of the bias of surface charges present on the
membrane (0 to -0.1 C m-2). If the action potential starts from a lower
resting potential (which can be down to -100 mV Hille (2001)), up to
$150\text{\,}\mathrm{\SIUnitSymbolMicro J}$ m-2 of heat is predicted by the
model. The predicted values fall in the range of experimental values (60 -
$180\text{\,}\mathrm{\SIUnitSymbolMicro J}$ m-2) Howarth  _et al._ (1979)).

##  IV Discussion

The purpose of this work was to provide a theoretical background for heat
production and absorption in neurons that goes beyond the simplistic analogy
with a capacitor. Based on an equilibrium thermodynamics description of the
membrane, its surface charges and double layers forming on each side, we
evaluated the amount of heat released due to free energy and entropy changes
during the action potential. The model assumes complete reversibility: all
internal energy changes are first converted into heat, and then all heat is
converted back to a change in the internal energy.

First, we found that the electric free energy of the cell membrane depends on
both the membrane potential and the transmembrane potential, as shown by the
new expression derived in this work:
${F}^{\mathrm{m}}_{\mathrm{el}}=\frac{1}{2}c_{\mathrm{m}}\>\phi_{\mathrm{t}}\>V_{\mathrm{m}}$.
In consequence, the amount of free energy stored by the membrane increases as
more surface charges are present on the internal versus the external side of
the membrane. Free energy changes in the diffuse layers surrounding the
membrane also increase significantly when a bias of surface charge is present,
however, these changes are offset by the entropy changes associated with the
polarization of water by the electric field, that are of comparable magnitude
but of opposite sign ($\sim$-120% at 0°C). Finally, entropy changes in the
membrane appear to be related to the dependence on temperature of the lipid
bilayer’s dimensions rather than of its dielectric permittivity. Recent
measurements of the dimensional variations allowed to determine that the
magnitude of entropy changes is comparable ($\sim$+82$\%$) to the one of free
energy changes.

The comparison of model predictions with the best-available measurements of
the heat production and absorption in neurons supports the idea that the heat
of nervous conduction has an electrical origin: assuming a typical action
potential that depolarizes the membrane from -70 mV to +20 mV and a surface
charge bias of $-0.05$ C m-2 between the two sides of the membrane, we predict
a production and absorption of heat of $\sim$60
$\text{\,}\mathrm{\SIUnitSymbolMicro J}$ m-2, which is sufficient to reach the
range of experimental values measured in the neurons of different organisms
(60 - 180 $\text{\,}\mathrm{\SIUnitSymbolMicro J}$ m-2) Howarth  _et al._
(1979). Should the size of the action potential be larger (starting from a
more negative resting potential), or the surface charge bias be more
pronounced, more heat would be evolved.

Clearly, our predictions have a critical dependency on the action potential
size and surface charge distribution between the internal and external sides
of the membrane. Both parameters can vary from one type of neuron to the
other, however, the values used for our calculations do not seem extravagant,
considering the range of values reported in literature for surface charge
density Hille (2001); Lakshminarayanaiah and Murayama (1975). The experimental
validation of our predictions requires to compare the heat production with the
true course of the action potential in neurons, which can only be obtained by
measuring the membrane potential intracellularly. With the recent development
of nano-electrodes Angle  _et al._ (2015); Duan  _et al._ (2013); Ferguson
_et al._ (2012), there is hope that it is becoming possible to record true
membrane potentials in the small neurons in which temperature changes can
effectively be measured. Although we cannot discard the possibility of other
sources of heat, such as heat arising from electrically-induced elastic
displacements of the membrane El Hady and Machta (2015); Lacoste  _et al._
(2009, 2007), our analysis suggests that the change in electrical energy of
the membrane represents the most prominent mechanism of heat production and
absorption by neurons during the propagation of the action potential.

Besides neurophysiology, the physics our work explores could be relevant for
engineering electrical double layer capacitors Janssen and van Roij (2017) or
designing capacitive deionization processes, which performance can be
influenced by the presence of surface charges, either immobile Biesheuvel  _et
al._ (2015) or modulated electrochemically He  _et al._ (2018). Moreover,
thermodynamic analyses in capacitive systems typically do not include the
entropy of dipoles in dielectric media Hemmatifar  _et al._ (2018). Here, we
showed that the entropy of dipoles brings a significant contribution to the
internal energy stored across the double layers and the associated heat
production and absorption.

##  V Author Information

Corresponding Authors: ∗E-mail<EMAIL_ADDRESS>
Author Contributions:
† A.D.L and J.P.D contributed equally to this work.

##  VI Acknowledgements

J.P.D. acknowledges support in the form of a National Science Foundation
Graduate Research Fellowship under Grant No. 1122374. A.D.L. acknowledges
support from the Louis-de-Broglie Foundation of the French Academy of
Sciences. The authors thank Mathijs Janssen and Maarten Biesheuvel for useful
comments.

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##  VII Appendix

###  VII.1 Fast equilibration of double layers

In this work, we describe ions and the electric potential in the diffuse
layers at equilibrium, with the Boltzmann distribution (Eqs. (9) and (10)).
This equilibrium assumption decouples the problem from time dependence for
relaxation of the double layers and simplifies it to a single dependence on
the membrane potential $V_{\mathrm{m}}$. However, in reality, $V_{\mathrm{m}}$
is time-dependent. We must thus verify that the double layers reach their
equilibrium conformation (ion concentration versus position) significantly
faster than the variation of $V_{\mathrm{m}}$ during the course of the action
potential. Two time scales can characterize the relaxation of diffuse layers:
diffusion $\tau_{D}={L^{2}}/{D}$ and charge density relaxation
$\tau_{L}={\lambda_{D}^{2}}/{D}$, where L is the characteristic length, D the
characteristic diffusivity, and $\lambda_{D}$ the Debye length. At low
voltages, Bazant  _et al._ (2004) have shown that the primary time scale for
diffuse charges dynamics in a non-Faradaic electrochemical cell is the
harmonic mean between these time constants, that is:
$\tau_{C}=\lambda_{D}\>L/D$. Interestingly, a recent study by Janssen (2019)
has shown that the easily obtainable $\tau_{C}$ decently describes double
layer relaxation also in cylindrical geometry, the relevant geometry for
neurons. By analogy with the electrochemical cell, the membrane can be seen as
sandwiched between two electrodes separated by a distance $L$, and on which a
difference of potential is applied Lacoste  _et al._ (2009); Ziebert and
Lacoste (2011).
In physiological conditions (ionic strength = 150 mM) and at 0$\degree$C, the
Debye length is less than 1 nm:

$\lambda_{D}=\sqrt{\frac{RT\>\varepsilon}{F^{2}\sum{z_{j}^{2}\>c_{j}}}}=\>0.6\>\mathrm{nm}.$
(28)

In the garfish olfactory nerve, one of the smallest axons studied, the
characteristic length is given by the radius of the cell, $L/2=0.25$
$\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Taking $D=10^{-9}$ m2 $s^{-1}$ we
find $\tau_{C}=0.3$ $\text{\,}\mathrm{\SIUnitSymbolMicro s}$. In the giant
squid axon, one of the largest axons ever studied, we have $L/2=200$
$\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and thus $\tau_{C}=240$
$\text{\,}\mathrm{\SIUnitSymbolMicro s}$. These characteristic times for
relaxation are small compared to the time scale of compound action potentials,
which is 10 to 100 ms Howarth  _et al._ (1979). While in the nonlinear regime
(where the double layer potential across each side is much larger than the
thermal voltage $\sim$24 mV) the time scale for bulk salt diffusion
$\tau_{D}=L^{2}/D$ can be important, the leading order dynamics occur at the
$\tau_{C}$ scale. Therefore, we neglect any double layer relaxation effect and
treat them at equilibrium.

###  VII.2 Fast equilibration of temperature inside the nerve

Another important assumption we make is the fast heat equilibration between
the cell membrane and the rest of the nerve. The validity of this assumption
can be verified based on scaling analysis on the heat equation, which yields a
characteristic time for heat diffusion:

$\tau_{\mathrm{h}}=\frac{\rho\>c_{\mathrm{p}}\>L^{2}}{K}$ (29)

where $\rho$ is the mass density, $c_{\mathrm{p}}$ the specific heat capacity,
$L$ the characteristic length and $K$ the thermal conductivity. With
$L=0.125\>$\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, $\rho=5.8$ g cm-3,
$c_{\mathrm{p}}=3.6$ J (g.K)-1 Howarth  _et al._ (1979), and using the thermal
conductivity of water at 0$\degree$C, $K=0.0056$ J (K cm s)-1 Weast  _et al._
(1988), we find $\tau_{\mathrm{h}}\sim 0.6$ µs. Thus, the heat equilibration
in the interior of the small nerve fibers in which the heat experiments were
conducted Howarth  _et al._ (1979) is on the order of microseconds, which is
much quicker than the timescale of the action potential in these nerves (10 to
100 ms) Howarth  _et al._ (1979). The nerve is solely composed of densely-
packed fibers, such that the mean distance between fibers is smaller than the
size of a single fiber Easton (1971). Thus, we assume that the whole nerve
volume is at thermal equilibrium during the action potential, and calculate
the rise of the nerve’s temperature as proportional to the amount of heat
dissipated:

$\Delta{T}=-\frac{\Delta{Q}}{c_{\mathrm{p}}}\>A_{\mathrm{m}},$ (30)

where $A_{\mathrm{m}}=65000$ cm2 g-1 is the total membrane area per mass of
nerve Howarth  _et al._ (1979).

###  VII.3 The Condenser Theory from Maxwell’s equation for Ampere’s law

Heat production and field energy can be unified by Maxwell’s equation for
Ampere’s law Jacobs (2013),

$E\>(\nabla\times{H})=E\>{J}+E\>{\frac{\partial{D}}{\partial{t}}},$ (31)

where $J$ is the current density (A m-2), $H$ is the magnetic field and $D$
the displacement. -$E\>{J}$ is the energy dissipated by an ionic current $J$
in an electric field $E$ as heat per unit of volume and time Jacobs (2013),
noted $\dot{Q}_{\mathrm{V}}$ (negative when heat is dissipated). Neglecting
magnetic contributions, and in one dimension, we find

$\dot{Q}_{\mathrm{V}}=-E\>J=E\>{\frac{\partial{D}}{\partial{t}}}.$ (32)

This relationship offers another way to understand the Condenser Theory:
during the action potential, positive charges (Na+ and K+ ions) move through
ion channels of the membrane Hille (2001), first in the same direction as the
electric field ($-E.J<0$, in one dimension), then against the electric field
($-E.J>0$), which results in heat production ($\dot{Q}_{\mathrm{V}}<0$) and
then heat absorption ($\dot{Q}_{\mathrm{V}}>0$). Let’s consider the situation
where the field is 0 at time zero and $E$ at time $t$. If the dielectric
medium is linear ($D=\varepsilon.E$), the heat absorbed between time $0$ and
$t$ is equal to the field energy at time $t$:

$\begin{split}Q_{\mathrm{V}}&=\int_{0}^{t}{E\>{\frac{\partial{D}}{\partial{t}}}}\>dt=\frac{1}{2}\int_{0}^{t}{\Big{(}E\>{\frac{\partial{D}}{\partial{t}}}+\frac{\partial{E}}{\partial{t}}\>{D}\Big{)}}\>dt\\\
&=\frac{1}{2}E\>{D}.\end{split}$ (33)

The second equality holds for any linear dielectric medium, whereas the third
is given by the fundamental theorem of calculus. Finally, integrating
$Q_{\mathrm{V}}$ over the membrane domain $\Omega_{\mathrm{M}}$ gives the heat
dissipated per unit of membrane surface area (J/m2):

$Q=\int^{+\infty}_{-\infty}{\int_{0}^{t}{E\>{\frac{\partial{D}}{\partial{t}}}}\>dt\>dx=\int^{+\infty}_{-\infty}\frac{1}{2}E\>D}\>dx.$
(34)

This last expression gives the amount of heat that is dissipated upon release
of the electric free energy of a dielectric medium Overbeek (1990); Jackson
(1999).

###  VII.4 Derivation of the electric free energy of the membrane capacitance
and of double layers

In this section we split the free energy of the $\Omega_{\mathrm{M}}$ domain
(given by Eq. (22) or (34)) into a contribution from the membrane capacitance
and one form the double layers. This separation is necessary to entropy
calculations (Section II.4). We start by expanding the field energy expression
over the internal diffuse layer, the membrane, and the external diffuse layer:

$\begin{split}F_{\mathrm{el}}=&\frac{1}{2}\bigg{[}\varepsilon\int_{-\infty}^{-\delta}{\bigg{(}\frac{d{\phi_{\mathrm{i}}(x)}}{d{x}}}\bigg{)}^{2}dx+\varepsilon_{\mathrm{m}}\int_{-\delta}^{0}{\bigg{(}\frac{d{\phi_{\mathrm{m}}(x)}}{d{x}}}\bigg{)}^{2}dx\\\
&+\varepsilon\int_{0}^{+\infty}{\bigg{(}\frac{d{\phi_{\mathrm{o}}(x)}}{d{x}}}\bigg{)}^{2}dx\bigg{]}.\\\
\end{split}$ (35)

Assuming the density of free charges to be zero at any point $]-\delta;0[$
inside the membrane, Poisson’s equation requires the field to be constant
across the membrane, such that $\frac{d{\phi_{\mathrm{m}}(x)}}{d{x}}$ =
$\frac{d{\phi_{\mathrm{m}}(0)}}{d{x}}$. The second term in the RHS of Eq.
(35), then simplifies to

$\begin{split}\varepsilon_{\mathrm{m}}\int_{-\delta}^{0}\bigg{(}\frac{d{\phi_{\mathrm{m}}(x)}}{d{x}}\bigg{)}^{2}dx&=\varepsilon_{\mathrm{m}}\int_{-\delta}^{0}\bigg{(}\frac{d\phi_{\mathrm{m}}(0)}{dx}\bigg{)}^{2}dx\\\
&=\varepsilon_{\mathrm{m}}\bigg{(}\frac{q}{\varepsilon_{\mathrm{m}}}\bigg{)}^{2}\int_{-\delta}^{0}{}dx\\\
&=\frac{q^{2}}{c_{\mathrm{m}}}.\\\ \end{split}$ (36)

The second equality is given by boundary conditions for the electric field,
Eq. (17). Next, integration by parts of the first and third terms in Eq. (35)
gives

$\begin{split}F_{\mathrm{el}}=&\frac{1}{2}\bigg{(}\varepsilon\>\left[{\frac{d{\phi_{\mathrm{i}}}}{d{x}}\>\phi_{\mathrm{i}}}\right]^{-\delta}_{-\infty}-\varepsilon\int_{-\infty}^{-\delta}{\phi_{\mathrm{i}}\>\frac{d^{2}{\phi_{\mathrm{i}}}}{d{x^{2}}}}dx+\frac{q^{2}}{c_{\mathrm{m}}}\\\
&+\varepsilon\>\left[{\frac{d{\phi_{\mathrm{o}}}}{d{x}}\>\phi_{\mathrm{o}}}\right]^{+\infty}_{0}-\varepsilon\int^{+\infty}_{0}{\phi_{\mathrm{o}}\>\frac{d^{2}{\phi_{\mathrm{o}}}}{d{x^{2}}}}dx\bigg{)}.\end{split}$
(37)

Remembering that the potential is constant at $-\infty$ and $+\infty$, using
Eqs. (13) and (14) and Poisson’s equation, Eq. (37) can be rearranged into

$\begin{split}F_{\mathrm{el}}=&\frac{1}{2}\bigg{(}\int_{-\infty}^{-\delta}{\rho_{\mathrm{i}}\>\phi_{\mathrm{i}}\>}dx+\Big{(}\sigma_{\mathrm{i}}+q\Big{)}\>\phi_{\mathrm{i}}(-\delta)\\\
&+\int^{+\infty}_{0}{\rho_{\mathrm{o}}\>\phi_{\mathrm{o}}\>}dx+\Big{(}\sigma_{\mathrm{o}}-q\Big{)}\>\phi_{\mathrm{o}}(0)+\frac{q^{2}}{c_{\mathrm{m}}}\bigg{)}.\\\
\end{split}$ (38)

As $q=-c_{\mathrm{m}}(\phi_{\mathrm{i}}(-\delta)-\phi_{\mathrm{o}}(0))$ by Eq.
(19), this last expression simplifies to

$\begin{split}F_{\mathrm{el}}=&\frac{1}{2}\bigg{(}\int_{-\infty}^{-\delta}{\rho_{\mathrm{i}}\>\phi_{\mathrm{i}}\>}dx+\sigma_{\mathrm{i}}\>\phi_{\mathrm{i}}(-\delta)\\\
&+\int^{+\infty}_{0}{\rho_{\mathrm{o}}\>\phi_{\mathrm{o}}\>}dx+\sigma_{\mathrm{o}}\>\phi_{\mathrm{o}}(0)\bigg{)}.\end{split}$
(39)

By applying the substitution
$\phi_{\mathrm{i}}=(\phi_{\mathrm{i}}-V_{\mathrm{m}})+V_{\mathrm{m}}$ and
using Eqs. (11) and (19), we separate the free energy in the diffuse layers
$(F^{\mathrm{DL}}_{\mathrm{el}})$ from the one in the membrane
$(F^{\mathrm{m}}_{\mathrm{el}})$:

$\begin{split}F^{\mathrm{DL}}_{\mathrm{el}}=&\frac{1}{2}\bigg{(}\int_{-\infty}^{-\delta}{\rho_{\mathrm{i}}(\phi_{\mathrm{i}}-V_{\mathrm{m}})}dx+\sigma_{\mathrm{i}}\>(\phi_{\mathrm{i}}(-\delta)-V_{\mathrm{m}})\\\
&+\int^{+\infty}_{0}{\rho_{\mathrm{o}}\>\phi_{\mathrm{o}}\>}dx+\sigma_{\mathrm{o}}\>\phi_{\mathrm{o}}(0)\bigg{)},\end{split}$
(40)

and

$F^{\mathrm{m}}_{\mathrm{el}}=\frac{1}{2}\>c_{\mathrm{m}}\>\phi_{\mathrm{t}}\>V_{\mathrm{m}},$
(41)

with
$F_{\mathrm{el}}=F^{\mathrm{DL}}_{\mathrm{el}}+F^{\mathrm{m}}_{\mathrm{el}}$.

###  VII.5 Derivation of entropy changes in the membrane

Within the Condenser Theory, it was suggested that entropy changes in the
lipid bilayer accompanying the action potential could account for up to 4
times the free energy changes Howarth  _et al._ (1979); Ritchie (1973). The
ratio between the entropy-related energy and the free energy was calculated as

$\frac{T\Delta{S}}{\Delta{F}}=\frac{T}{C}\>\frac{\partial{C}}{\partial{T}}$
(42)

where $C$ is the membrane capacitance. However, no full derivation of this
equation is reported in the literature. Here we will show under which
conditions Eq. 42 can be derived, based on the demonstration Frohlich (1968).
For simplicity, we will consider a membrane without any surface charges,
holding a charge $q$ on each side (here $q$ is the capacitive charge in C),
across a potential difference $V$. This simplification will ease notations
without modifying the relation between the membrane’s entropy and electric
free energy studied here. We start by considering the change of internal
energy $dU$ (in J) following an increment of charge $dq$. The first law of
thermodynamics reads

$\displaystyle dU$ $\displaystyle=dQ+dW$ $\displaystyle=dQ+V\>dq$

where $dQ$ is the influx of heat and $dW$ is the electrical work done on the
membrane. Note that the electrical work did not appear explicitly in the
thermodynamic system defined in Section II.2, since no electrical work is done
on $\Omega_{\mathrm{M}}$ by the surroundings (the latter is however implicitly
accounted for as $\Delta{F_{\mathrm{el}}}$ in Eq. 21). For the purpose of this
demonstration, the thermodynamic system considered here consists of the
membrane only ($x\>\epsilon\>]-\delta;0[$), which enables us to express the
electrical work associated with the depolarization and repolarization of the
membrane explicitly. Assuming that the capacitance $C$ remains constant with
$V$ but depends on the temperature $T$, we obtain

$dq=d(C.V)=dC.V+C.dV=\frac{\partial{C}}{\partial{T}}\>dT.V+C.d(V).$ (43)

Thus, a variation of $q$ may be due to a variation of temperature or of
potential, the capacitance being assumed to remain constant for the range of
physiological potentials ($ca.$ \- 100 mV to + 20 mV). By taking $T$ and
$V^{2}$ as independent variables, the first law now becomes

$dQ+V^{2}\frac{\partial{C}}{\partial{T}}\>dT+\frac{1}{2}C\>d(V^{2})=dU=\frac{\partial{U}}{\partial{T}}\>dT+\frac{\partial{U}}{\partial{(V^{2})}}\>d(V^{2}).$
(44)

Note that $dQ$ is not a total differential, however $dS=dQ/T$ is one, for a
reversible process Jackson (1999). This property means that a unique function
$S(T,V^{2})$ must exist, such that

$dS=\frac{\partial{S}}{\partial{T}}\>dT+\frac{\partial{S}}{\partial{(V^{2})}}\>d(V^{2}).$
(45)

Thus if there is a relation $dS=f(T,V^{2}).dT+g(T,V^{2}).dV^{2}$, where f and
g are two unknown functions, the condition that $dS$ is a total differential
requires that

$\frac{\partial{f}}{\partial{(V^{2})}}=\frac{\partial{g}}{\partial{T}},$ (46)

as both sides of Eq. 46 are equal to
${\partial^{2}{S}}/({\partial{T}\partial{(V^{2})}})$, by Eq. 45. Next, by
substituting for $dS=dQ/T$ in Eq. (44), we find

$dS=\frac{1}{T}\Big{(}\frac{\partial{U}}{\partial{T}}-V^{2}\frac{\partial{C}}{\partial{T}}\Big{)}dT+\frac{1}{T}\Big{(}\frac{\partial{U}}{\partial{(V^{2})}}-\frac{C}{2}\Big{)}d(V^{2}).$
(47)

This equation is analogous to Eq. (45) and therefore, Eq. (46) becomes

$\frac{\partial}{\partial{(V^{2})}}\Big{\\{}\frac{1}{T}\Big{(}\frac{\partial{U}}{\partial{T}}-V^{2}\frac{\partial{C}}{\partial{T}}\Big{)}\Big{\\}}=\frac{\partial}{\partial{T}}\Big{\\{}\frac{1}{T}\Big{(}\frac{\partial{U}}{\partial{(V^{2})}}-\frac{C}{2}\Big{)}\Big{\\}}.$
(48)

By carrying out differentiations we find (take ${\partial{C}}/{\partial{T}}$
as a constant):

$\frac{\partial{U}}{\partial{(V^{2})}}=\frac{C}{2}+\frac{1}{2}\frac{\partial{C}}{\partial{T}}T$
(49)

Finally, upon integration, we obtain

$\Delta{U}=\frac{1}{2}\int^{V^{2}}_{0}\Big{(}C+\frac{\partial{C}}{\partial{T}}T\Big{)}d(V^{2})=\frac{1}{2}CV^{2}+\frac{1}{2}\frac{\partial{C}}{\partial{T}}T\>V^{2}.$
(50)

The first term is the free energy change $\Delta{F}$ of the capacitor, whereas
the second is the energy associated with a change of entropy $T\Delta{S}$.
Thus we have

$T\Delta{S}=\Delta{U}-\Delta{F}=\frac{1}{2}\frac{\partial{C}}{\partial{T}}T\>V^{2}=\frac{T}{C}\>\frac{\partial{C}}{\partial{T}}\Delta{F},$
(51)

and Eq. (26) holds, under the condition that the membrane capacitance has a
linear dependence on temperature, but is constant with potential. A number of
recent studies indicate that there exists such a linear dependence and that
the value of ${\partial{C}}/{\partial{T}}$ is $\sim 0.3\%/\degree$C for a wide
range of cell membranes Plaksin  _et al._ (2018); Carvalho-de Souza  _et al._
(2015); Martino  _et al._ (2015); Hotka and Zahradnik (2014); Shapiro  _et
al._ (2012). Why does the membrane capacitance change with temperature? Using
the classical model for the membrane capacitance,
$C=\varepsilon_{\mathrm{m}}\>{A}/{d}$, where $A$ is the membrane area, and
$\delta$ the membrane thickness, the temperature dependence could be
attributed to a variation of the dielectric permittivity
${\partial{\varepsilon_{\mathrm{m}}}}/{\partial{T}}$, or to a variation of the
membrane dimensions (${\partial{A}}/{\partial{T}}$ and
${\partial{\delta}}/{\partial{T}}$). Dimensional changes have been measured in
artificial lipid bilayers Pan  _et al._ (2008); Norbert Kucerka and Katsaras
(2011); Szekely  _et al._ (2011), and in neurons, these changes correspond to
a $0.11$+-$0.03\%$/K reduction in the lipid membrane thickness and a
$0.22$+-$0.03\%/\degree$C increase in the area per lipid, resulting in a
linear increase of the capacitance with temperature of 0.3 $\%$/K Plaksin  _et
al._ (2018). Little is known on how the membrane’s dielectric permittivity
varies with temperature, however
${\partial{\varepsilon_{\mathrm{m}}}}/{\partial{T}}$ has been measured for
unsaturated fatty acids (the molecules composing the core of the membrane),
yielding values more than one order of magnitude smaller than the combined
dimensional changes Plaksin  _et al._ (2018); Lizhi  _et al._ (2008).

###  VII.6 Supplementary plots

Figure 6: Free energy changes in the membrane, according to the revised
Condenser Theory (this work, $\Delta F^{\mathrm{m}}_{\mathrm{el}}$ ) and to
the formula suggested by Howarth  _et al._ (1979) and Ritchie and Keynes
(1985) ($\Delta F^{\mathrm{m}}_{\mathrm{el,Ritchie}}$ ). Strikingly, the
latter overestimates free energy changes. Surface charge densities assumed are
$\sigma_{\mathrm{i}}=-0.1$ C m-2 and $\sigma_{\mathrm{o}}=-0.05$ C m-2. Figure
7: Profiles of the electric potential across the membrane and the surrounding
diffuse layers, for different values of the membrane potential
$V_{\mathrm{m}}$ (-70 to +30 mV). TOP: surface charge densities are equal on
each side ($\sigma_{\mathrm{i}}=\sigma_{\mathrm{o}}=-0.05$ C m-2). BOTTOM: a
bias of surface charges makes the internal side more negative
($\sigma_{\mathrm{i}}=$-0.1 C m-2 and $\sigma_{\mathrm{o}}=-0.05$ C m-2), such
that the transmembrane potential $\phi_{\mathrm{t}}$ need not to reverse when
the membrane potential is positive (in purple).