Autapse-induced logical resonance in the FitzHugh–Nagumo neuron

It was demonstrated that the chaos-driven FitzHugh–Nagumo (FHN) neuron can be considered as a logic system to implement the reliable logical operations through the mechanism of logical resonance. Autapse (meaning the self-synapse) widely exists in various kinds of neurons, and it significantly affects the neuronal dynamics and functionalities. However, the effects of autapse on logical resonance have not been reported yet. Here, we explore the effects of autapse on the reliability of AND & NAND logical operations based on the autaptic FHN neuron model with time-varying coupling intensity. The numerical results demonstrate that there are the optimal ranges of parameters (including autaptic time delay, amplitude, frequency and phase fluctuation of autaptic coupling intensity) at which the reliability of logical operations can be maximized. Namely, autapse-induced logical resonance can be realized in the autaptic FHN neuron model. More interestingly, multiple logical resonances can be obtained by regulating autaptic time delay, phase fluctuation of autaptic coupling intensity, as well as frequency ratio between and autaptic coupling intensity and external periodic driving force. Finally, an intuitive interpretation for autapse-induced logical resonance is given based on the motion of the particle in the potential landscape.


Introduction
Signal transmissions and information exchanges between neurons depend on the key functional structures called synapses. If a neuron connects with itself, then an autapse forms. This special kind of synapses is firstly named by Van der Loos and Glaser [1]. Thereafter, autapse has been experimentally verified to widely exist in different brain regions of various species [2,3]. Functionally, autapse provides a timedelayed self-feedback mechanism, and thus, autapse is often expressed mathematically as a time-delayed feedback term in the original neuron models [4]. On this basis, numerous theoretical studies have explored the effects of autaptic time delay and conductance on the neuronal dynamics and functionalities.
Autapse can modulate electrical activities [5][6][7]. For instance, an excitatory autapse enhances firing, while an inhibitory autapse suppresses firing. Additionally, transitions of firing patterns can be induced by autapse [8,9] and different modes of firing can be selected through autapse. Some biological functionalities of neuronal networks are dependent on collective electrical behaviors of neurons, and collective electrical behaviors are originated from synchronization. It was demonstrated that autapse can excite and regulate collective electrical behaviors of neurons via generating stable regular waves, such as target waves and spiral waves [10,11]. In fact, synchronization, wave emitting and propagation induced by autapse can be observed in various neuronal networks [11][12][13]. However, increasing autaptic coupling intensity may reduce the firing rate of neurons and thus destroy synchronization of firing patterns in neuronal networks [14,15]. When synchronized and asynchronized subpopulations coexist in a network, this collective behavior is called chimera state, which is strongly related to quite a few neuronal processes [16]. It was reported that autapse has effect on chimera state in the Hindmarsh-Rose neuronal network [17]. Moreover, mode-locking behaviors may occur when neurons are subjected to periodic force, and mode-locking firing patterns play positive roles in neural information coding. Autapse can enhance or suppress the modelocking firing of Hodgkin-Huxley neuron [18]. Modelocking firing is sensitive to autaptic time delay so that autapse can be considered as a potential control strategy to adjust the mode-locking firing behaviors [18].
The phenomenon that noise induces enhancement of weak signal in the nonlinear systems is named stochastic resonance (SR). Noise-induced resonance dynamics can be significantly affected by autapse [19]. More specifically, multiple SRs can be induced by appropriately adjusting autaptic time delay when autaptic coupling strength is in a specific range [19]. Moreover, autapse can also induce inverse SR in which noise inhibits average firing rate of neurons [20]. Noise-induced oscillations are rather irregular for small or large noise amplitudes, but coherent oscillations can be generated for moderate noise amplitudes. This phenomenon is referred to as coherence resonance (CR) [21]. Yilmaz et al. reported that autapse-induced multiple CRs can be observed in a neuron and neuronal network through appropriately tuning autaptic conductance levels and autaptic time delay [22]. Instead of stochastic noise, chaotic activities in deterministic nonlinear systems can also amplify weak signal or enhance performance of weak signal detection via chaotic resonance. In the appropriate range of parameters, autapse significantly increases the chaotic resonance [23]. Via resonance mechanism, autapse can also improve informationprocessing capacity of neuron, such as enhancing the efficiency of signal transmission [24] and the propagation of pacemaker rhythm across the whole network [25].
In 2009, Murali and coworkers extended the classical concept of SR to the logic areas and achieved the reliable logical operations in a bistable system under the optimal range of noise intensity [26]. This phenomenon is named logical stochastic resonance (LSR) [26]. In fact, stochastic noise  or some other non-stochastic elements such as periodic force [48][49][50][51][52], chaotic force [53][54][55][56][57], time delay [58][59][60][61], parameter [62] and coupling [63,64] can be used to improve the reliability of logic systems even for subthreshold inputs. Logical resonance gives a potential option to construct the new-style logical elements. This new type of logical elements may be energy saving since they can work reliably under the subthreshold inputs. Thus, it has aroused wide concern and has a promising prospect in energy-efficient computing building modules of futuristic computational systems. We have recently proposed the concept of logical chaotic resonance (LCR) based on a bistable system [53]. And LCR has been successfully extended to the two-state excitable Fitzhugh-Nagumo (FHN) neuron model [56]. Many previous studies have shown that autapse significantly affects the firing patterns and the resonance dynamics of neurons. However, there is no related reports about whether autapse affects logical resonance in the neuron. For this purpose, we will explore the effects of autapse on logical operations based on the autaptic FHN neuron model.

Model and scheme
Our analysis is based on the following autaptic Fitzhugh-Nagumo neuron model [21,56,65] Timescale ratio e is fixed to 0.01 so that all motions can be separated into the fast (only x changes) and slow (y % x À x 3 =3) ones [21]. Bifurcation parameter a is set to 1.05 for AND logical operation and 1 for NAND logical operation. For a j j [ 1, the system has a stable fixed point and is excitable. Namely, small perturbations may bring the system to produce large excursion. Due to this highly nonlinear response to perturbations, the dynamics of the forced system may be non-trivial. For a j j\1, the system has a limit cycle consisting of two pieces of slow motion connected with fast jumps [21]. B sin xt ð Þ denotes a periodic driving force with amplitude B and angular frequency x. s 1 and s 2 denote two independent aperiodic twolevel square waves, respectively, and can encode two independent logic inputs according to the predefined encoding scheme. In consequence, external input signals s 1 and s 2 are modulated by the periodic force B sin xt ð Þ. Parameter d signifies a bias, and it takes value 0 for AND logic gate and -0.6 for NAND logic gate. Through this paper, B ¼ 0:15 and x ¼ 10 À0:5 . Consequently, when a ¼ 1:05, external periodic driving force B sin xt ð Þ is unable to evoke any spikes in the absence of autaptic current I aut . Autaptic delayed stimulus is expressed as below [66] Here s denotes autaptic time delay due to finite propagation speed during axonal transmission. c refers to autaptic coupling intensity and is time-varying according to the following formula [24,67]: Here g and x represent the amplitude and angular frequency of autaptic coupling intensity c, respectively. N signifies the frequency ratio between autaptic coupling intensity c and external periodic driving force B sin xt ð Þ. W(t) is the standard Wiener process, while r indicates the intensity of the Wiener process. W(t) is used here to model stochastic fluctuation of phase in autaptic coupling intensity c. When r is equal to 0, autaptic coupling intensity c is time-periodic. Clearly, r controls the degree of deviation of c from regular time-periodic process, as displayed in Fig. 1. As previously mentioned, the generation of W(t) is based on [40,68,69] v 1 and v 2 are two independent random numbers, which are uniformly distributed in the interval 0-1. The initial value of W(t) takes a random value in the interval 0-1. The dynamical equation is integrated using the Euler's method with a fixed time step Dt ¼ 0:001 time units.
Note that s 1 and s 2 can encode two binary inputs: They are considered as logic 1 when their values are fixed to 0.3 or logic 0 when their values are fixed to 0. Therefore, four logic input sets can be produced by combining two binary inputs (Table 1). In addition, the FHN neuron is a two-state system; thus, the responses of the system to external stimuli can be mapped into logic outputs 1 and 0 (Table 1) based on the state of the FHN neuron. Hereafter, we can define a success probability P to characterize the reliability of logic gates [26,28].

P ¼
The number of successful runs The total number of runs ð5Þ under five representative intensities r of the Wiener process W(t). From bottom to up, as r is increased, c shows more and more deviation from regular time-periodic process. g = 0.1 and N = 1 For all possible logic input sets, if the logic output of the system is always correct in a run according to logic input-output associations (i.e., the truth table) ( Table 1), then this run is regarded as the successful one. Because there exists time delay between the input and the output of the system, failures happened in the transient process (defined as the first 10% of time period, i.e., 200 time units) must be tolerated when we calculate P. For the same reason, s 1 and s 2 must be required to remain unchanged during this time period. In order to ensure statistical validity, the total number of runs is set as 1000 and calculation time window is set as 10,000 time units. Note that control runs with larger total number of runs, longer calculation time window and smaller time step reveal no significant difference in the main results.

Autapse-induced logical resonance for AND logical operation
We firstly consider the case that autaptic coupling intensity c is time-periodic, and has the same frequency x with external periodic driving force B sin xt ð Þ. As shown in Fig. 2, when s ¼ 0:5, there is no spike at all, and thus, the FHN neuron is in the resting state for all possible value of s 1 þ s 2 . As a consequence, the logic output of the system is always 0 for all possible logic input sets according to the truth table (Table 1). When s ¼ 2, if the value of s 1 þ s 2 is 0.6 (i.e., the logic input set is (1, 1)), spike can be evoked, and thus, the FHN neuron is in the excited state, the response of the system can be interpreted as logic output 1. The FHN neuron is in the resting state (i.e., the logic output of system is 0) if the value of s 1 þ s 2 is 0 or 0.3 (i.e., the logic input set is (1, 0), (0, 1) or (0, 0)). Consequently, associations between logic input and output of the system are always consistent with the truth table for AND logic gate (Table 1). That is to say, this run is a successful one. When s ¼ 5, it can be observed clearly that there are spikes occasionally (i.e., the logic output is 1) if the value of s 1 þ s 2 is 0.3 (i.e., the logic input set is (1, 0) or (0, 1)). In this case, the logic input-output associations do not match the required one for AND logic gate (Table 1) so that the desired logic output is not reliable. The above-mentioned results show that when the autaptic time delay s is moderate, neither small nor large, the consistent AND logic output can be obtained in the autaptic FHN neuron. Likewise, the similar phenomena can be observed clearly for three representative amplitudes g of autaptic coupling intensity c (Fig. 3). Specifically, moderate g, neither small nor large, yields a consistent AND logic output. Taken together, when s and g take some certain optimal values, the reliable AND logical operation can be realized in the forced autaptic FHN neuron.
In order to search the optimal ranges of parameters, and to quantify the reliability of logical operations, dependence of success probability P on increasing autaptic time delay s and amplitude g are plotted in Fig. 4. Distinctly, there exists an optimal range of autaptic time delay s at which success probability P of obtaining specific logical operation can be maximized (Fig. 4a). Moreover, from top to bottom, as g is decreased, the optimal window of s moves toward right (i.e., the direction of larger s). On the whole, for smaller g, larger s is required to obtain reliable logical operation. More interestingly, for specific g (such 0.015), multiple logical resonances can be observed clearly (Fig. 4a). Likewise, we can see the optimal range of g at which AND logical operation is reliable (Fig. 4b). The optimal window of g moves slightly toward right with the decrease in the autaptic time Table 1 Truth table for  delay s, and the optimal window of g becomes significantly wider (Fig. 4b). These results can be further confirmed via the contour plot of P in g-s plane (Fig. 5). The red region in Fig. 5 denotes the optimal ranges of g and s where logical resonance occurs. Next, we investigate the effects of frequency ratio log 10 N ð Þ and phase fluctuation intensity r on logical resonance. As shown in Fig. 6, when r ¼ 0, moderate frequency ratio can induce a consistent AND logic output. In other words, for all possible logic input sets, the desired logic output is always achieved, and thus, AND logical operation is reliable (Fig. 6). For small or large frequency ratio log 10 N ð Þ, the autaptic FHN neuron is in the resting state, the responses of the system to input signals s 1 and s 2 do not match the logic input-output associations ( Table 1). The desired logic operation is unrealizable (Fig. 6). In a word, there exists the optimal window of frequency ratio between periodic driving force and time-varying autaptic coupling intensity c. As mentioned above, r can be used to represent the degree of phase fluctuation in autaptic coupling intensity c (Fig. 1). When r ¼ 0, there is no spike except the transient process. The logic output is not correct when the logic input set is (1, 1) (i.e., s 1 þ s 2 is equal to 0) (top panel Fig. 7). However, slight phase fluctuation (i.e., small r) can lead to the correct logic response of the system for all logic input sets (middle panel in Fig. 7). Although further increasing r decreases the firing rate (i.e., the number of spikes is divided by calculation time window), it does not destroy the reliability of logic gate, and the desired logic results are always obtained for all possible logic input sets (Fig. 7). Further, the curves of success probability P vs. log 10 N ð Þ and r are drawn in Fig. 8. The resonant peaks can be observed by adjusting frequency ratio log 10 N ð Þ between autaptic coupling intensity c and external periodic driving force B sin xt ð Þ (Fig. 8a). In addition, when r is the relatively small, multiple logical resonances can be carried out by regulating frequency ratio (Fig. 8a). The width of the optimal range of frequency ratio log 10 N ð Þ increases with increasing r (Fig. 8a). For large frequency ratio (such as log 10 N ð Þ ¼ À2 or 2:5), the logical operation is not reliable at small r, but the logical operation becomes reliable at large r (Fig. 8b). For a certain specific frequency ratio (such as Fig. 4 Dependence of success probability P on increasing a autaptic time delay s and b amplitude g of autaptic coupling intensity c. N = 1 and r ¼ 0: Distinctly, there exist the optimal ranges of s and g at which success probability P of obtaining AND logical operation can be maximized. In addition, multiple logical resonances can be observed in (a) for specific g (such as 0.015)  , the logical operation is always reliable (Fig. 8b). Interestingly, for log 10 N ð Þ ¼ 0, double logical resonances can be induced by regulating r, and moderate phase fluctuation (moderate r) destroys the reliability of logical operation (Fig. 8b).

Autapse-induced logical resonance for NAND logical operation
In this section, we explore the possibility of realizing NAND logical operation using the same system. As displayed in Fig. 9, for small or large autaptic time delay s, NAND logical operation is not reliable. Specifically, when s ¼ 1, if the value of s 1 þ s 2 is 0.3 (i.e., the logic input set is (1, 0) or (0, 1)), the desired logic output is 1 according to the truth table for NAND logic gate (Table 1). However, the forced FHN cannot firing in this case, that is, the logic output of the system is 0 (Fig. 9). When s ¼ 3:2, we can see some spikes (i.e., the logic output is 1) if s 1 þ s 2 takes value 0.6 (i.e., the logic input set is (1,1)). Thus, patterns of input-to-output mapping do not match the required one (Table 1). For moderate autaptic time delay s (such as s ¼ 2), the logic output of the system is always correct for all logic input sets (Fig. 9). Likewise, under the moderate amplitude g of autaptic coupling intensity c, the reliable NAND logical operation can be implemented, as shown in Fig. 10. Clearly, the reliable NAND logical operation can be achieved at some optimal frequency ratios log 10 N ð Þ (Fig. 11). The above results demonstrate the occurrence of autapse-induced logical resonance for NAND logical operation. As indicated by Fig. 12a, multiple logical resonances can be induced by regulating autaptic time delay s, especially for small amplitude g of autaptic coupling intensity c. In addition, with the increase of g, the optimal window of s shrinks in size (Fig. 12a). At the moderate s, the width of the optimal window of g can be maximized (Fig. 12b). Additionally, there are the optimal windows of frequency ratios log 10 N ð Þ to obtain reliable NAND logical operation (Fig. 12c). Increasing r can enhance success probability P when log 10 N ð Þ is relatively small (Fig. 12c).

An intuitive interpretation for autapse-induced logical resonance
We firstly consider the stimuli-free FHN model that there are no autaptic delayed stimulus (i.e., c = 0) and external stimuli (consisting of periodic driving force B sin xt ð Þ, bias d, input signals s 1 and s 2 ). Thus, Eq. (1) becomes In this case, the system has a unique fixed point at (x 0 ¼ Àa; y 0 ¼ a 3 =3 À a). Let t ¼ eT in Eq. (6), then one obtains [70,71] Fig. 8 Dependence of success probability P on a log 10 N ð Þ and b r. s ¼ 0:5, g ¼ 0:05. a There are optimal ranges of log 10 N ð Þ at which success probability P of obtaining the desired logical results can be maximized. b Appropriate phase fluctuation r yields a consistent AND logic output where U is a potential function [70,71]: Due to e ( 1, y can be approximated as a constant on the timescale from Eq. (7). Thus, y may be treated as a parameter in Eq. (8), as previously mentioned [70]. At a fixed y (such as -0.66), U is a double-well Fig. 11 Sampled time series of x, y and s 1 þ s 2 under three representative frequency ratios log 10 N ð Þ. s ¼ 2, g ¼ 0:02 and r ¼ 0: Clearly, top and bottom panels yield a consistent NAND logic output Fig. 12 Dependence of success probability P on increasing a autaptic time delay s, b amplitude g and c frequency ratio log 10 N ð Þ. a, b N = 1 and r ¼ 0: c s ¼ 2, g ¼ 0:02. Distinctly, there are the optimal ranges of s, g and log 10 N ð Þ at which success probability P of obtaining NAND logical operation can be maximized. In addition, multiple logical resonances can be observed in a for some specific g (such as 0.005 and 0.01) potential with two minima located at the values of x (the resting state) and the spike value (Fig. 13a).
Adjusting y can alter the height of potential barrier ( Fig. 13(a)). In particular, the double-well potential can become a single-well one (Fig. 13a). In addition, for a fixed y, the height of potential barrier is related to the value of a [70,71]. Then, the potential landscape for the stimuli-free FHN model (i.e., Equation (6)) can be constructed by changing x and y (Fig. 13b). It is worth noting that in the presence of autaptic delayed stimulus the potential function can be rewritten to include the delay term [72], V ¼ U x; y ð Þþcx 2 =2 Àcxx s . Here x s denotes the delayed state. It can immediately be seen that the potential landscape becomes dependent on autaptic coupling intensity c and the delayed state x s at the earlier time.
Despite this, we can still use the forced motion of the particle to give an intuitive interpretation for autapse-induced logical resonance (Fig. 13b). As illustrated in Fig. 14a, when the value of s 1 þ s 2 is 0 (i.e., the logic input set is (0, 0)), due to ðs 1 þ s 2 þ dÞB sin xt ð Þ and d = 0 for AND logical operation, without external stimuli ðs 1 þ s 2 þ dÞB sin xt ð Þ (consisting of periodic driving force B sin xt ð Þ, bias d, input signals s 1 and s 2 ), the particle always lies in the left potential well (i.e., the resting state), indicated by the blue triangle (Fig. 14a). Thus, the logic output of the system is 0 in this case. When the value of s 1 þ s 2 is 0.3 (i.e., the logic input set is (1, 0) or (0, 1)), the external stimuli (s 1 þ s 2 þ dÞB sin xt ð Þ, together with self-feedback gain of the system, have not enough energy to hop the particle over the potential barrier to the right potential well (i.e., the excited state). Thus, the particle just slightly oscillates around the resting state, indicated by the green loop (Fig. 14a). Consequently, the logic output is also 0 in this case. But when the value of s 1 þ s 2 is 0.6 (i.e., the logic input set is (1, 1)), the external stimuli (s 1 þ s 2 þ dÞB sin xt ð Þ and self-feedback gain of the system have enough energy to hop the particle over the potential barrier to the right potential well, indicated by the black squares (Fig. 14a). Therefore, the logic output of the system is 1. According to the truth table (Table 1), the reliable AND logical operation can be implemented. Namely, autapse-induced logical resonance can occur in the appropriate ranges of parameters.
The interpretation for NAND logical operation is similar to one for AND logical operation. Note that it becomes easier for the particle to roll into the positive potential well due to the decrease of height of potential barrier caused by decreasing a (a ¼ 1 for NAND logical operation). Specifically, when the value of s 1 þ s 2 is 0.6 (i.e., the logic input set is (1, 1)), which is corresponding to the case that s 1 þ s 2 is equal to 0 for AND logical operation, there is no external stimuli ðs 1 þ s 2 þ dÞB sin xt ð Þ in this case, and the particle always lies in the left potential well (i.e., the resting state), indicated by the blue loop (Fig. 14b). As a result, the logic output of the system is 0. When the value of s 1 þ s 2 is 0.3 (i.e., the logic input set is (1, 0) Fig. 13 Potential U(x, y) for the stimuli-free FHN model (i.e., Equation (6)). a dependence of U on x for some fixed y; b dependence of U on x and y or (0, 1)) or 0 (i.e., the logic input set is (0, 0), the external stimuli ðs 1 þ s 2 þ dÞB sin xt ð Þ lets the particle roll into the positive potential well, indicated by the green and black points (Fig. 14b). Therefore, according to the truth table (Table 1), the reliable NAND logical operation can be implemented in the optimal windows of parameters. What is noteworthy is that the potential landscape is dependent on the delayed effect and time-varying coupling intensity of autapse. This lets the motion of the particle near the resting state becomes very different (Fig. 15). Because of this, the complex potential landscape makes the system may have more than one the optimal windows of parameters to realize the reliable logical operations, and thus, autapse-induced multiple logical resonances are emerged.

Conclusions
In this paper, autaptic coupling intensity c is regarded as a time-varying one. Then, the effects of autapse on logical resonance are investigated based on the autaptic FHN neuron model. The success probability P of obtaining specific logical operations is used to measure the reliability of logical operations. The first case is that autaptic coupling intensity c is a timeperiodic one which has the same frequency with subthreshold external periodic driving force. The numerical results have demonstrated that in this case there are the optimal ranges of autaptic time delay s, amplitude g of autaptic coupling intensity c where AND & NAND logical operations are always reliable. Namely, autapse-induced logical resonance can be achieved by regulating s or g. Interestingly, for a certain specific g, autaptic time delay s can induce multiple logical resonances. The second case is that autaptic coupling intensity c is an aperiodic one with phase fluctuation and its frequency is N times of external periodic driving force. In the second case, we have explored the effects of frequency ratio N and phase fluctuation r on the reliability of AND & NAND logical operations. The numerical results have manifested that varying log 10 N ð Þ or r can cause the reliability of logical operations to be maximized. In other words, autapse-induced logical resonance can be implemented by regulating frequency ratio log 10 N ð Þ or the degree of phase fluctuation r. In addition, for a certain specific log 10 N ð Þ or r, multiple logical resonances can be obtained by adjusting r or s log 10 N ð Þ. Taken together, the autaptic FHN neuron with timevarying coupling intensity can be regarded as a logic system, and the reliability of logical operations can be significantly influenced by autapse.
Some limitations of this study must be considered. The neuronal model used in this study is very simple although the FHN model serves as a representative example of excitable systems, and is widely used in different fields [21], Further, autapse can be of two types: electrical synapse and chemical synapse. Only electrical autapse is investigated here. It deserves further study how chemical autapse affects logical resonance in the more biologically realistic model of neurons, such as Hodgkin-Huxley model. As a simple paradigm, we have only shown the possibility of using excitable neuronal systems with autapse to mimic AND & NAND Boolean logic gates with conventional 2-input logic gate structure. Note that the other fundamental operations, such as OR, NOR and XOR logic gates can be constructed by combining NAND or NOR operations [73]. In addition, coupling conventional 2-input logic gates may lead to more complexity in actual experimental circuit realizations [73]. Thus, multiple-input logic gates have apparent advantage in reducing complexity of circuit implementations. It still needs further research how to implement all logic gates, memory latch and multiple-input logic gates in the same model.
As research on artificial intelligence (AI) booms, many neuron-inspired or brain-inspired neuromorphic devices have been developed to improving the computational capacity at the lowest possible cost [74]. How to implement energy-efficient neuron circuits is one of the main challenges in the field of neuromorphic computing [75]. What is noteworthy is that Ferroelectric field-effect transistor (FeFET)based circuit has been successfully applied to mimic FitzHugh-Nagumo neuron and the proposed neuron implementation is very energy-efficient [76]. Even for the subthreshold inputs, the autaptic FHN neuron can still operate reliably as a logic gate through the mechanism of logical resonance. Consequently, this new-style morphable logic gate shows promise in reducing energy consumption. Although several limitations remain, the results obtained here might facilitate the development of neuron-inspired logic devices using the mechanism of logical resonance in actual systems.
Funding Not applicable.
Data availability The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Declarations
Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.