Energy Model of Neuron Activation

LETTER Communicated by Lubomir Kostal

Yuriy Romanyshyn [email protected] Lviv Polytechnic National University, Lviv 79013, Ukraine, and University of Warmia and Mazury in Olsztyn, Olsztyn 10-719, Poland

Andriy Smerdov [email protected] Poltava State Agrarian Academy, Poltava 36003, Ukraine

Svitlana Petrytska [email protected] Lviv Polytechnic National University, Lviv 79013, Ukraine

On the basis of the neurophysiological strength-duration (amplitude- duration) curve of neuron activation (which relates the threshold amplitude of a rectangular current pulse of neuron activation to the pulse duration), as well as with the use of activation energy constraint (the threshold curve corresponds to the energy threshold of neuron activation by a rectangular current pulse), an energy model of neuron activation by a single current pulse has been constructed. The constructed model of activation, which determines its spectral properties, is a bandpass filter. Under the condition of minimum-phase feature of the neuron activation model, on the basis of Hilbert transform, the possibilities of phase- frequency response calculation from its amplitude-frequency response have been considered. Approximation to the amplitude-frequency response by the response of the Butterworth filter of the first order, as well as obtaining the pulse response corresponding to this approximation, give us the possibility of analyzing the efficiency of activating current pulses of various shapes, including analysis in accordance with the energy constraint.

1 Introduction

Models of neurons (including models of their activations) can be divided into two large categories: static and dynamic. Static models are usually used for the formation of artiﬁcial neural networks of the ﬁrst and second generations (with threshold and continuous functions of activation) and are described by the corresponding activation functions of the following kinds: threshold, linear, linear with saturation, sigmoidal, and others (Haykin,

Neural Computation 29, 502–518 (2017) c 2017 Massachusetts Institute of Technology doi:10.1162/NECO_a_00913

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 Energy Model of Neuron Activation 503

1998). Dynamic models (Gerstner & Kistler, 2002) represent dynamic processes in a neuron and are used in both neurophysiological research and the artificial neural networks of the third generation (Maass, 1997). It is possible to distinguish the following models among the neuron dynamic ones: • Integrate-and-fire (accumulation-and-discharge) neuron model. This neuron model (Gerstner & Kistler, 2002; Abbott, 1999) is one of most widely used models in computing neuroscience and is an alternative to the complicated models for solving such tasks in which the shape of a neural pulse has no essential importance. The time ratios for series of consecutive pulses, however, are significant during neuron activation, including synchronization tasks (Bressloff & Coombes, 1997; Coombes & Bressloff, 1999). • Integrate-and-fire neuron model with adaptation. In this model (Dayan & Abbott, 2001), an adaptation is introduced—a jump-like increase in the threshold of activation with every pulse of activation, which leads to gradual reduction of the frequency of pulses that are generated (spike-frequency adaptation). • Gerstner-Kistler model. This model (spike response model) (Gerstner & Kistler, 2002) determines a simplified dynamics of a neuron, to inputs of which neural pulses from other neurons come. It is applied in analyses of neural dynamics and neural coding. • Hodgkin-Huxley model (Hodgkin & Huxley, 1952; Gerstner & Kistler, 2002) is one of the most important models in neuroscience and represents a system of four nonlinear differential equations of the first order in four unknown variables. To mathematically analyze the Hodgkin- Huxley equations system is a rather complicated problem in the research of neural structures with a large number of neurons. As a result, there are modifications and simplifications of this model, for example, the FitzHugh-Nagumo two-dimensional model (FitzHugh, 1961; Gerstner & Kistler, 2002). The expediency in using these models, as well as some others, for modeling processes in the cerebral cortex is analyzed in Izhikevich (2004). In the course of the construction of all these models, issues of the energy of activation of a neuron were not directly considered, though energy constraint is significant in neurophysiological processes. A considerable number of publications have studied the energy factor in research on the processes in neurons and bioneural structures. In particular Yu and Liu (2013) investigated the energy efficiency of neural systems in the process of pulse signal transfer by means of action potentials. The optimal number of ion-conducting channels for maximizing energy efficiency for a single neuron has been discovered on the basis of computer simulation of the Hodgkin-Huxley stochastic version model (Yu & Liu, 2013). The energy efficiency of the neuron array has been also studied. Wang and Wang

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 504 Y. Romanyshyn, A. Smerdov, and S. Petrytska

(2014) explored neuron coding in relation to the energy that is consumed by neurons, as well as correlation between energy expenditures and neural network parameters (number of neurons, number of connections, delays of transmission). Kostal, Lansky, and McDonnel (2013) examined methods of information theory for information transmission analysis in an empirical neuronal model. They found that information transmission is related to the metabolic cost of neuronal activity and determined the optimal ra- tio of amount of information to metabolic cost. Kostal and Lansky (2013) extended those results for populations of neurons. Kostal and Shinomoto (2016) recently investigated the information capacity of Poisson neurons under metabolic cost constraints. Sengupta, Laughlin, & Niven (2013), on the basis of analysis and comparison of three regimes of activation of a stochastically modified Hodgkin-Huxley neuron model, determined the optimal energy efficiency and efficiency of an information coding regime. Attwell and Laughlin (2001) conducted on energy analysis of processes of neuron activation at the level of export and import of Na+ and K+ ions through a membrane. Using some approximations enabled them to obtain estimations of energy expenditures of input current of activation for the formation of a single potential of action. In spite of all this work, the interrelation between the energy of neuron activation and the known threshold strength-duration curve has not been considered. In this letter, we set out a model of neuron activation constructed from the energy point of view in aggregate with the strength-duration curve for a rectangular current pulse of activation.

2 Construction of the Neuron Activation Model

To construct the neuron activation model, we used the known experimental neurophysiological dependence between the threshold value of the amplitude of a rectangular current pulse of activation and its duration, which we call the strength-duration curve. (Boinagrov, Loudin, & Palanker, 2010; Geddes & Bourland, 1985; Mogyoros, Kiernan, Burke, & Bostock, 1998; Mogyoros, Kiernan, & Burke, 1996). This dependence represents the reac- tion of a neuron to a single rectangular current pulse of activation. If the pulse parameters are below the threshold values, neuron activation does not occur; if the threshold values are reached or exceeded, an activation of the neuron and formation of a neural pulse occur. To describe the created neural pulse, an approximation to the experimental neural pulse can be used. For mathematical approximation to this curve, various formulas are used, among which the following are known: • Hyperbolic (Lapicque’s equation), (τ ) = ( + τ /τ ) I I0 1 0 (2.1)

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 Energy Model of Neuron Activation 505

Figure 1: Normalized strength-duration curves, which relate the threshold amplitude of a rectangular current pulse of neuron activation to the pulse duration, calculated for the Hodgkin-Huxley model; hyperbolic approxima- (τ ) = ( + τ /τ ) tion, I I0 1 0 ; generalized hyperbolic (power form) approximation (τ ) = + (τ /τ )α α = α = . I I0[1 0 ], at 2and 0 5; and exponentially hyperbolic ap- (τ ) = − (−τ/τ ) −1 proximation, I I0[1 exp 0 ] .

• Generalized hyperbolic (power form)

(τ ) = + (τ /τ )α ,α> I I0[1 0 ] 0 (2.2) • Exponentially hyperbolic dependences (Tuckwell, 1988a),

(τ ) = − (−τ/τ ) −1 I I0[1 exp 0 ] (2.3)

where I(τ ) is the threshold amplitude of the rectangular current pulse, τ τ is the pulse duration, and I0 and 0 are constants (rheobase and chronaxia, respectively). In Figure 1, four approximations to the normalized strength-duration curve are given. In the same figure, the strength-duration curve, which is calculated for the Hodgkin-Huxley model, is presented. The values of parameters (Gerstner & Kistler, 2002) are as follows: maximal values of = 2 specific conductances of ion-conducting channels are gK 36 mS/cm , = 2 = . 2 gNa 120 mS/cm ,andgL 0 3mS/cm; the specific capacitance of the

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 506 Y. Romanyshyn, A. Smerdov, and S. Petrytska

membrane is C = 1 μF/cm2; and the shifted values of the basic voltages (at = =− which the rest potential is equal to zero) are VNa 115 mV, VK 12 mV, = . and VL 10 6 mV. For this curve, the numerical values of rheobase are ≈ . μ 2 τ ≈ . I0 2 24 A/cm and of chronaxia 0 1 66 ms. Analogical curves for the Hodgkin-Huxley model are presented in Tuckwell (1988b). It is clear from Figure 1 that the hyperbolic approximation, equation 2.1, is the closest to the strength-duration curve for the Hodgkin-Huxley model. In this connection, it is the approximation that should be used to construct an energy model of neuron activation. It is also the simplest one from the point of view of its use in further transformations. To construct the model, the approximation 2.1 was used. In addition, the energy constraint—that is, the threshold strength-duration curve of activation of the model by a rectangular current pulse corresponds to the energy threshold of activation—was used. On the basis of these two conditions, we constructed the model of neuron activation. To substantiate the latter condition, we calculate the dependence of the energy of input rectangular current pulses on the duration of the pulses in the case of threshold activation for the Hodgkin-Huxley model of neuron. This energy is determined by 2(τ ) τ = Cu + 3 ( ( ) − ) + 4( ( ) − ) E I [gNam h u t VNa gKn u t VK 2 0 + ( ( ) − ) , gL u t VL ]dt (2.4)

where u(t) is the voltage on the neuron membrane; τ is the duration of the pulse of threshold activation; I is the corresponding density of input current; and m, h, n are the known parameters of the Hodgkin-Huxley model (Gerstner & Kistler, 2002). τ/τ The dependence of this energy on the normalized duration 0 of the τ/τ pulse is depicted in Figure 2. Under the variation of 0 from 0.3 to 1 in Figure 2, the change in the input energy does not exceed 15%. This allows us to consider the approximate magnitude of the threshold energy as a constant value. The energy of the input rectangular current pulse of threshold of neuron activation is not of a constant value if the amplitude of the pulse and its duration are related between each other by the approximation 2.1 to the strength-duration curve. Because of this, the need to construct such an element (energy model of activation) at the output of which the energy of ) the signal takes a constant value (energy threshold of activation Eth for an input signal of the rectangular shape whose amplitude and duration are related by the approximation 2.1 arises. It is also expedient to make a transition from a time-dependent to a frequency-dependent representation of signals because the energy spectra (ω) (ω) Win at the input and Wout at the output of the model are linearly

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 Energy Model of Neuron Activation 507

Figure 2: Dependence of the energy of pulses of threshold activation on their normalized duration for the Hodgkin-Huxley model of neuron, which demon- strates weak dependence of this energy (the change in the input energy does τ/τ not exceed 15% under the variation of 0 from 0.3 to 1), which allows us to approximately consider the magnitude of the threshold energy as a constant value.

related to each other by the frequency-dependent response R(ω) in terms of power. The spectral density of energy (energy spectrum) of the input rectangular current pulse with the amplitude I and the duration τ at the model input is determined by

4I2 ωτ W (ω) = sin2 . (2.5) in ω2 2

The energy spectrum of signal at the output of the activation model is determined by

(ω) = (ω) (ω). Wout R Win (2.6)

The energy of the output signal is determined by

∞ 4I2 R(ω) ωτ E = sin2 dω. (2.7) out π ω2 0 2

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 508 Y. Romanyshyn, A. Smerdov, and S. Petrytska

Therefore, with an amplitude and duration of the activating current pulse, related by the dependence 2.1, the output energy remains constant:

= . Eout Eth (2.8)

With the help of our model of neuron activation, we suggest the transition from the activation threshold in terms of the amplitude of the activating current pulse (which is represented by strength-duration curve) to a threshold in terms of energy; that is, we use a single physical quantity instead of functional dependence. As a result, we obtained the integral equation

∞ ∞ 2I2(τ ) R(ω) R(ω) dω − cos ωτ dω = E . (2.9) π ω2 ω2 th 0 0

(ω)/ω2 ∈ ;∞) To solve this equation, we assume that the function R L1[0 , that is, it is absolutely integrable in the interval [0;∞). Then the second term on the left side of equation 2.9 is proportional to the inverse Fourier cosine transform of the function R(ω)/ω2, for which the following equality is true:

∞ R(ω) lim cos ωτ dω = 0. (2.10) τ→∞ ω2 0

Taking into account that

(τ ) = , τlim→∞ I I0 (2.11)

we obtain

∞ 2I2 R(ω) 0 dω = E . (2.12) π ω2 th 0

As a result, the integral equation, 2.9, takes the form

∞ 2 R(ω) E τ (2τ + τ ) cos ωτ dω = th · 0 0 . (2.13) π ω2 2 (τ + τ )2 0 I0 0

Using Fourier cosine transform, we get

∞ E τ (2τ + τ ) R(ω) = ω2 th 0 0 cos ωτ dτ. (2.14) 2 (τ + τ )2 I0 0 0

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 Energy Model of Neuron Activation 509

The integral in this expression can be obtained in analytical form. As a result of the corresponding transformations (Fikhtengolts, 1966), we derive the expression

E R(ω) = th x2 2g(x) − 1 + xf(x) , (2.15) 2τ I0 0 = ωτ ( ) =− ( ) − ( ) ( ) = ( ) − ( ) where x 0; g x ci x cos x si x sin x; f x ci x sin x si x cos x; and

∞ cos t ci(x) =− dt − integral cosine; (x > 0), x t ∞ sin t si(x) =− dt − integral sine,(x ≥ 0). x t

Let us determine the asymptotic behavior of the function R(ω) for ω → 0 and ω →∞.Forsi(x) and ci(x), the following power series expansions are true (Korn & Korn, 1961):

π 1 x3 1 x5 si(x) =− + x − + − ..., 2 3! 3 5! 5 1 x2 1 x4 ci(x) =C + ln x − + − ..., (2.16) 2! 2 4! 4

where C = 0, 577216 ...is the Euler-Mascheroni constant. Taking into account the expansions of sin x and cos x into power series, we obtain the asymptotic representation of R(ω) for ω → 0:

2E R (ω) =− th x2 ln x [1 + o(x)] . (2.17) ω→ 2τ 0 I0 0

For the functions g(x) and f (x) when ω →∞, the following expansions are acceptable (Fikhtengolts, 1966): 1 1 3! − (2m − 1)! g(x) = − + ...+ (−1)m 1 + ... , x x x3 x2m−1 1 2! − (2m − 2)! f (x) = 1 − + ...+ (−1)m 1 + ... . (2.18) x x2 x2m−2

As a result, we obtain the asymptotic representation R(ω) for ω →∞: 12E 1 1 R (ω) = th + o . (2.19) ω→∞ 2τ 2 3 I0 0 x x

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 510 Y. Romanyshyn, A. Smerdov, and S. Petrytska

Figure 3: Amplitude-frequency responses of neuron activation model, calculated for the exact solution of the model and on the basis of approximation to the amplitude-frequency response of the ﬁrst-order ﬁlter.

As a consequence, we obtain

lim R(ω) = lim R(ω) = 0. (2.20) ω→0 ω→∞

From here, it follows that the neuron activation model, which determines its spectral properties, is represented by a bandpass ﬁlter. The frequency response (in accordance with power) for expression 2.15 is represented in a normalized form in Figure 3 for the values of normalized frequency within two decades (from 0.1 to 10). The approximated amplitude-frequency response, which is obtained in section 4, is also given here. The central frequency and pass band of the frequency response are determined by the ﬁlter parameters. Bandpass properties of the neuron model, taking into account noise, have been noted in Plesser (1998).

3 Possibilities of Determination of the Phase-Frequency Response of the Neuron Activation Model

Though from the point of view of the constructed energy model of neuron activation, only the amplitude-frequency (energy) response of the ﬁlter,

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 Energy Model of Neuron Activation 511

which unambiguously determines transformation of signal energy, has sig- nificance. In order to completely identify the filter (i.e., to ensure the possibility to determine the shape of the output pulse), it is also necessary to have the phase-frequency response in addition to the amplitude-frequency response of the signal transmission. For its unambiguous determination on the basis of the amplitude-frequency response, it is essential to assume that the filter is a minimum-phase element, that is, its frequency response must have no zero on the right half of the complex s-plane (D’Azzo, Houpis, & Sheldon, 2003). In this case, the modulus and the phase of frequency response of the minimum-phase circuit are functionally related by Hilbert transform (Whitaker & Benson, 2004). All passive ladder circuits (i.e., those are configured purely on the basis of series and parallel connections) are minimum-phase ones (Davis & Agarwal, 2001). Exactly such a ladder structure has, for example, a Hodgkin-Huxley neuron model, with signal transmission from input to output carried out in only one way. This establishes that the constructed model of neuron activation is a minimum-phase element. The assumption of minimum-phase properties is grounded and used, for example, in Krumin, Shimron, and Shoham (2010), in the course of construction and in identification of the linear-exponential-Poisson model of a neuron. To determine the phase-frequency response of the filter in the neuron model, two main methods are acceptable:

1. Direct calculation of Hilbert transform after previous transformation of the corresponding expressions 2. The use of the presentation of Hilbert transform in a spectral area and the Fourier transform

For minimum-phase systems, the modulus |K( jω)| and the phase ϕ(ω) of frequency-dependent response are functionally related by the Hilbert inverse transform, ∞ ∞ 1 ln K( jz) 1 lnW(z) ϕ(ω) =− V.p. dz =− V.p. dz, (3.1) π −∞ ω − z 2π −∞ ω − z

where V.p. stands for the principal value of the improper integral. Since it is impossible to obtain analytical expressions for the corresponding indeﬁnite integral, there is a need for numerical determination of its principal value; such determination is connected with difﬁculties in computing. In this regard, there is a need to reduce this integral to a convergent one (which could be integrated numerically) by means of the corresponding transformations.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 512 Y. Romanyshyn, A. Smerdov, and S. Petrytska

Preliminary transformations of the integral are associated with its divi- sion into two integrals: ∞ ( ) ∞ ( ) ϕ(ω) ∼ 1 lnW z − 1 lnW z . π − ω dz π + ω dz (3.2) 2 0 z 2 0 z

After the change of variables (in the ﬁrst integral, z − ω = y;inthe second integral, z + ω = y) and the apportionment of singular points, we obtain ω ( + ω) ϕ(ω) ∼ 1 1 W y + π ln (− + ω)dy 2 0 y W y ∞ 1 1 W(y + ω) + ln dy. (3.3) 2π ω y W(y − ω)

v = y−ω After one more substitution y+ω in both integrals, we obtain 1 2ω 2ω 1 W +v W −v 1 ϕ(ω) = ln 1 1 · dv. (3.4) π 2ωv 2ωv 1 − v2 0 W 1+v W 1−v

In the derived integral, the limits of integration are ﬁnite; singular points coincide with these limits. In addition, the obtained improper integral is already convergent. This follows from asymptotic expressions for the integrand at integration limits. All of this provides the possibility of imple- menting the following procedure for the integral calculation. The segment ε ε − ε − ε [0; 1] is divided into three parts: [0; 1], [ 1 and 1 2]; and [1 2;1].The ε ε values 1 and 2 are chosen according to the conditions of the possible use of asymptotic approximations for a integrand near the values 0 and 1. On ε − ε the segment [ 1;1 2], the integral is calculated by means of one of the known methods of numerical integration (e.g., the Simpson method) and ε − ε on the segments [0; 1]and[1 2; 1] by direct integration of approximated functions. The calculation of the phase-frequency response by the Hilbert transform can also be implemented in the spectral representation on the basis of the Fourier transform. As the integral on the right side of expression 3.1 is | ( ω)| − 1 determined by the convolution of the functions ln K j and πω,the spectrum of the phase-frequency response is equal to product of the spectra,

( ξ)= ( ξ)· ( ξ), Sϕ j S1 j S2 j (3.5)

( ξ) | ( ω)| ( ξ) where S1 j is the spectrum of the function ln K j and S2 j is the − 1 spectrum of the function πω.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 Energy Model of Neuron Activation 513

( ξ) The spectrum S2 j is ⎧ ⎪ − ,ξ< ∞ ⎨ j 0 ( ξ)=− 1 − jξω ω = ,ξ= . S2 j e d 0 0 (3.6) −∞ πω ⎩⎪ j,ξ>0

( ξ) | ( ω)| Thus, having the spectrum S1 j of the function ln K j ,itispossi- ble to obtain the spectrum Sϕ ( jξ). Then the phase-frequency response is determined by the inverse Fourier transform of the spectrum Sϕ ( jξ):

∞ 1 jωξ ϕ(ω) = Sϕ ( jξ)e dξ. (3.7) 2π −∞

Both methods are cumbersome from a computing standpoint; therefore, it is expedient to approximate the amplitude-frequency response, and from there on, it is easier to obtain the phase-frequency response.

4 Approximation to Amplitude-Frequency Response of Filter in the Neuron Model

Rather complicated analytical expressions in equation 2.15 signiﬁcantly complicate the use of the model, especially while determining the shape of the output pulse for a given input one. With this, all transformations are to be carried out numerically. Besides, the expression obtained for the amplitude-frequency response, not a fractionally rational expression rel- ative to ω, does not give us the possibility of directly using the known results of the circuit theory and obtaining the circuit implementation of the mathematical model. In this regard, it is necessary to approximate the amplitude-frequency response. As far as the obtained neuron activation model represents a bandpass ﬁlter, in order to approximate its frequency response (in terms of power), we use the following expression,

˜ 2 A2x2 K( jω) = , (4.1) ( 2 + 2 )( 2 + 2 ) x D1 x D2

| ˜( ω)|2 (ω) = ωτ where K j is the approximation value of R ; x 0;andA, D1, D2 are constants. This approximation corresponds to the Butterworth simplest function of the ﬁrst order, which presents the frequency response of a low-frequency

prototype. Values of the parameters A, D1,andD2 are determined from the conditions of the coincidence of the values of exact and approximated

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 514 Y. Romanyshyn, A. Smerdov, and S. Petrytska

Figure 4: Phase-frequency response of the ﬁlter in an approximated model of neuron activation obtained on the basis of an assumption about the minimum- phase nature of the model.

curves at the points of maximum and at the levels of 0.5 from the maximum value. In this situation, the obtained values of the parameters of the ≈ . ≈ . ≈ . approximating expression are A 5 55, D1 5 01, and D2 0 54. The approximating curve is presented in Figure 3, its greatest deviation from the calculated one, in accordance with formula 2.15, which takes place in the ˜ interval (0; 0.4), while R(0) =|K(0)|2 = 0. Determination of the phase-frequency response for a fractionally rational expression of the frequency response in terms of power becomes consid- erably simpler. In this, the known results of circuit theory are used, and the use of the Hilbert transform becomes unnecessary. The frequency response in complex form is obtained as a result of simple transformations of frequency response in terms of power. As a result we obtain,

ωτ ˜( ω) = j 0A . K j ( ωτ + )( ωτ + ) (4.2) j 0 D1 j 0 D2

The phase-frequency response of the energy model of neuron activation, which is calculated according to the obtained expression, is presented in Figure 4. The shape of this response corresponds to the phase response of

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 Energy Model of Neuron Activation 515

+π/ ωτ = a first-order bandpass filter with the phase change from 2at 0 0 −π/ ωτ →∞ to 2 for 0 . The zero value of the phase corresponds to the normalized frequency 1.64. Let us determine the maximum and the central frequency of the approximated frequency response of the filter. The square of the modulus of frequency response is represented as ˜ A2 x2 x2 |K( jω)|2 = − . (4.3) 2 − 2 2 + 2 2 + 2 D2 D1 x D1 x D2

The necessary condition of an extremum of the square of the response modulus is | ˜( ω)|2 2 ( 2 + 2 ) − 3 ( 2 + 2 ) − 3 d K j = A 2x x D1 2x − 2x x D2 2x 2 − 2 ( 2 + 2 )2 ( 2 + 2 )2 dx D2 D1 x D1 x D2 2A2x (D − D )(x2 − D D ) = 1 2 1 2 = 0. (4.4) 2 − 2 ( 2 + 2 )2( 2 + 2 )2 D2 D1 x D1 x D2

ω We obtain the central frequency 0 and the maximum value of the modulus of the frequency-dependent response: ω τ = ;|˜( ω)| = /( + ). 0 0 D1D2 K j max A D1 D2 (4.5)

On the basis of the derived expression for frequency response, it is easy to obtain the pulse response, which is determined by the inverse Fourier transform and is represented in analytical form: ( ) = A −D1t − −D2t ; ≥ . g t τ ( − ) D1 exp τ D2 exp τ t 0 0 D1 D2 0 0 (4.6)

˜ Considering that |K(0)|=0, we have

∞ g(t)dt = 0, (4.7) 0

that is, the areas of the positive and negative parts of the pulse response are equal. This response in the normalized form is presented in Figure 5.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 516 Y. Romanyshyn, A. Smerdov, and S. Petrytska

Figure 5: Normalized pulse response of the neuron activation model—response to δ-impulse.

5Conclusion

In the known dynamic models of a neuron, including those of its activation, the relationship between the known strength-duration (amplitude- duration) curve (the threshold amplitude of a rectangular activation current pulse versus its duration) and the energy threshold of neuron activation is not considered. The energy constraint of activation, which consists of transforming the threshold strength-duration curve for a rectangular current pulse of activation into an energy threshold of activation, leads to the indication that the neuron activation possesses a bandpass feature. The obtained frequency-dependent response of the activation model in terms of power can be approximated by that of the Butterworth filter of the first order. Having supposed that the neuron activation model possesses the minimum-phase feature, its phase-frequency response can be determined from the amplitude-frequency one on the basis of the Hilbert transform. The constructed model of neuron activation can be used to research the efficiency of activating current pulses of various shapes, as well as for statements and solving the problem of determinating the shape of an energy-optimum activating pulse.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 Energy Model of Neuron Activation 517

References

Abbott, L. F. (1999). Lapicque’s introduction of the integrate-and-fire model neuron (1907). Brain Research Bull., 50, 303–304. Attwell, D., & Laughlin, S. B. (2001). An energy budget for signaling in the grey matter of the brain. Journal of Cerebral Blood Flow and Metabolism, 21, 1133–1145. Boinagrov, D., Loudin, J., & Palanker, D. (2010). Strength-duration relationship for extracellular neural stimulation: Numerical and analytical models. Journal of Neu- rophysiology, 104, 2236–2248. Bressloff, P. C., & Coombes, S. (1997). Synchrony in an array of integrate-and-fire neurons with dendritic structure. Phys.Rev.Lett., 78, 4665–4668. Coombes, S., & Bressloff, P.C. (1999). Mode-locking and Arnold tongues in integrate- and-fire neural oscillators. Physical Review E, 60, 2086–2096. Davis, W. A., & Agarwal, K. (2001). Radio frequency circuit design. New York: Wiley. Dayan, P., & Abbott, L. F. (2001). Theoretical neuroscience. Computational and mathematical modeling of neural systems. Cambridge, MA: MIT Press. D’Azzo, J. J., Houpis, C. H., & Sheldon, S. N. (2003). Linear control system analysis and design with MATLAB (5th ed.). New York: Marcel Dekker. Fikhtengolts, G. M. (1966). Course of differential and Integral calculus. Moscow: Nauka Publishing, Fizmalit (in Russian). FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1, 445–466. Geddes, L. A., & Bourland, J. D. (1985). The strength-duration curve. IEEE Trans. on Biomedical Engineering, 32, 458–459. Gerstner, W., & Kistler, W. M. (2002). Spiking neuron models: Single neurons, populations, plasticity. Cambridge: Cambridge University Press. Haykin, S. (1998). Neural networks: A comprehensive foundation (2nd ed.). Upper Saddle River, NJ: Prentice Hall. Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and application to conduction and excitation in nerve. Journal of Physiology, 117, 500–544. Izhikevich, E. M. (2004). Which model to use for cortical spiking neurons? IEEE Trans. on Neural Networks, 15, 1063–1070. Korn, G. A., & Korn, T. M. (1961). Mathematical handbook for scientists and engineers. New York: McGraw-Hill. Kostal, L., & Lansky, P. (2013). Information capacity and its approximations under metabolic cost in a simple homogeneous population of neurons. Biosystems, 112 (3), 265–275. Kostal, L., Lansky, P., & McDonnel, M. D. (2013). Metabolic cost of neuronal information in an empirical stimulus-response model. Biol. Cybern., 107, 355–365. Kostal, L., & Shinomoto, S. (2016). Efficient information transfer by Poisson neurons. Mathematical Biosciences and Engineering, 13 (3), 509–520. Krumin, M., Shimron, A., & Shoham, S. (2010). Correlation-distortion based identification of linear-nonlinear-Poisson models. Journal of Computational Neuroscience, 29 (1–2), 301–308. Maass, W. (1997). Networks of spiking neurons: The third generation of neural network models. Neural Networks, 10, 1659–1671.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021 518 Y. Romanyshyn, A. Smerdov, and S. Petrytska

Mogyoros, I., Kiernan, M. C., & Burke, D. (1996). Strength-duration properties of human peripheral nerve. Brain, 119, 439–447. Mogyoros, I., Kiernan, M. C., Burke, D., & Bostock, H. (1998). Strength-duration properties of sensory and motor axons in amyotrophic lateral sclerosis. Brain, 121, 851–859. Plesser, H. E. (1998). Noise turns integrate-fire neuron into band-pass filter. (Gottingen¨ Neurobiology Report II, 760). Gottingen:¨ Max Planck Institute. Sengupta, B., Laughlin, S. B., & Niven, J. E. (2013). Balanced excitatory and inhibitory synaptic currents promote efficient coding and metabolic efficiency. PLoS Comput. Biol., 9 (10), e1003263. Tuckwell, H. C. (1988a). Introduction to theoretical neurobiology, Vol. 1: Linear cable theory and dendritic structure. Cambridge: Cambridge University Press. Tuckwell, H. C. (1988b). Introduction to theoretical neurobiology, Vol. 2: Nonlinear and stochastic theories. Cambridge: Cambridge University Press. Wang, Z., & Wang, R. (2014). Energy distribution property and energy coding of a structural neural network. Frontiers in Computational Neuroscience, 8. Whitaker, J., & Benson, B. (2004). Standard handbook of audio engineering. New York: McGraw-Hill. Yu, L., & Liu, L. (2013). The optimal size of stochastic Hodgkin-Huxley neuronal systems for maximal energy efficiency in coding of pulse signals. arXiv:1308.4122v1 [q-bio.NC]

Received March 24, 2015; accepted August 30, 2016.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/NECO_a_00913 by guest on 27 September 2021