Method for determining optimal temperature regulation and control interval in whole growth stage of greenhouse pepper based on curvature method

Abstract

The invention discloses a curvature method-based method for determining an optimal temperature regulation interval in a whole growth stage of greenhouse peppers, which comprises the following steps of: measuring photosynthetic rates of the peppers in different growth stages under different environmental conditions through a nested experiment, and constructing a data set; the environmental conditions comprise temperature, CO2 concentration and photosynthetic photon flux density; taking the growth stage, the CO2 concentration, the temperature and the photosynthetic photon flux density as input vectors, constructing a photosynthetic rate prediction model by adopting a back propagation neural network optimized by a genetic algorithm, and generating a corresponding photosynthetic rate response surface; dispersing the photosynthetic rate response surface into discrete points, calculating the Gaussian curvature of each point, and constructing a Gaussian curvature response surface; and discretizing the Gaussian curvature response curved surface into a temperature-Gaussian curvature response curve, taking a u chord value as a measure, determining a critical boundary of the response curve by adopting a random restart hill climbing method, and finally obtaining an optimal temperature regulation and control interval.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of agricultural intelligence, and particularly relates to a method for determining an optimal temperature regulation interval of a greenhouse pepper in a whole growth stage based on a curvature method. BACKGROUND

[0002] In facility production, extreme temperatures in the greenhouse can cause flower and fruit drop, developmental retardation, and other damage to crops. Traditional greenhouse temperature regulation mainly relies on fixed threshold methods or seeks maximum photosynthetic rate (Pn) to ensure optimal yield. However, the former imposes strict restrictions on temperature through threshold values, resulting in delayed responses and a lack of alignment with the physiological needs of crops. The latter can temporarily increase productivity, but maintaining continuous Pn peaks requires a large amount of energy consumption, leading to an imbalance between facility energy expenditure and growth benefits. This difference emphasizes the urgent need to transition to dynamic range adjustment strategies based on crop physiological feedback. Studies have shown that the dynamic changes in Pn, as a core metabolic process controlling the growth stage of crops, can accurately quantify the effects of temperature stress in real time, providing a feasible solution to the above challenges. Notably, recent research has identified a flexible region for Pn regulation, where crop productivity remains stable when Pn decreases by about 20%. This finding provides a theoretical basis for the development of dynamic range control strategies.

[0003] Based on this flexible region theory, previous studies have demonstrated its applicability in various regulatory scenarios, such as irrigation and supplemental lighting. The results show that allowing Pn to vary within a moderate range not only ensures stable production but also significantly reduces energy consumption for temperature regulation. In this context, a Pn prediction model is introduced as the basis for establishing dynamic temperature regulation standards. Accurate identification of the temperature inflection point in the photosynthesis model is crucial for optimizing regulation efficiency and maximizing crop growth.

[0004] Recently, curvature theory has been widely used to find inflection points, as it can describe the degree of curvature of a curve or surface at a specific point. Gao et al. constructed a Pn surface under the interaction of light and CO2 and used the u-string length algorithm to calculate the maximum curvature point of the discrete curve. This method can explore the target inflection point of light and CO2 co-regulation. However, although the u-string algorithm has robust anti-rotation and anti-noise properties, making it an ideal choice for two-dimensional feature point calculation, it cannot fully capture the three-dimensional morphology in space. SUMMARY

[0005] To overcome the shortcomings of the above prior art, the purpose of the present application is to provide a method for determining an optimal temperature regulation interval of a greenhouse pepper in a whole growth stage based on a curvature method, to more accurately divide the temperature regulation interval of crops in multiple growth stages.

[0006] In order to achieve the above object, the technical scheme adopted by the present application is:

[0007] A greenhouse pepper optimal temperature regulation interval determination method based on curvature method, comprising the following steps:

[0008] Step 1, through nested experiment, the photosynthetic rate of pepper in different growth stages under different environmental conditions is measured, and a data set is constructed; the environmental conditions include temperature, CO2 concentration and photosynthetic photon flux density;

[0009] Step 2, taking growth stage, CO2 concentration, temperature and photosynthetic photon flux density as input vector, adopting genetic algorithm optimized back propagation neural network to construct photosynthetic rate prediction model, and generating corresponding photosynthetic rate response surface;

[0010] Step 3, the photosynthetic rate response surface is discretized into discrete points, the Gaussian curvature at each point is calculated, and the Gaussian curvature response surface is constructed;

[0011] Step 4, the Gaussian curvature response surface is discretized into temperature-Gaussian curvature response curve, taking u chord value as the measurement, adopting random restart hill climbing method to determine the critical boundary of the response curve, and obtaining the optimal temperature regulation interval.

[0012] In one embodiment, the growth stage includes seedling stage, flowering stage, pre-fruit stage, mid-fruit stage and post-fruit stage, and the different environmental conditions are setting several temperature gradients, several photosynthetic photon flux density gradients and several CO2 concentration gradients.

[0013] In one embodiment, the data obtained by nested experiment is normalized, and the formula is as follows:

[0014]

[0015] Wherein, x i ′ is the normalized data, x i is the measured data under the i th environmental condition, is the minimum value of the measured data, is the maximum value of the measured data.

[0016] In one embodiment, the back propagation neural network has a three-layer feedforward structure, including an input layer, a single hidden layer and an output layer, and the back propagation method using mean square error as loss function is used to train the model, and the calculation process is as follows:

[0017]

[0018] Wherein, f(X) is the photosynthetic rate prediction value, w ij is the weight value between the hidden layer and the input layer, w jkb is the threshold value between the hidden layer and the input layer j b is the threshold value between the hidden layer and the input layer k b is the threshold value between the hidden layer and the input layer, k is the number of input variables, n is the data volume, h(·) is the Sigmoid activation function, which compresses the output to the interval [0, 1], g(·) is the ReLU transfer function, is the t-th feature under the i-th environmental condition, t = 1, 2, 3, 4, representing growth stage, CO2 concentration, temperature and photosynthetic photon flux density respectively.

[0019] In one embodiment, the input layer has 4 nodes corresponding to the four input variables of growth stage, CO2 concentration, temperature and photosynthetic photon flux density, the hidden layer has 10 nodes, and the output layer has 1 node, outputting the predicted value of photosynthetic rate.

[0020] In one embodiment, step 2, a photosynthetic rate response surface is constructed with photosynthetic photon flux density and temperature as x and y coordinates; step 3, photosynthetic photon flux density, temperature and photosynthetic rate are normalized to solve the significant amplitude difference between parameters.

[0021] In one embodiment, step 3, any discrete point of the photosynthetic rate response surface is taken as point P, and there are four adjacent base points around point P, corresponding to the positions of its adjacent faces. The geometry approximates point P to be infinitely close to the base surface. Point P and its four surrounding base points form adjacent triangles, and the formula is as follows:

[0022]

[0023]

[0024] In the formula, N is the number of triangles formed by point P and adjacent base points, A is the sum of the areas of all triangles, and the triangle formed by point P, point v i and point v i+1 is taken as an example, θ i represents the vertex angle of the i-th triangle, and the side lengths of the triangle are denoted as l i , l i+1 and k i , s = (l i +l i+1 +k i ) / 2 represents the half-perimeter of the triangle intersecting point P; if the local surface area around point P is flat, i.e. point P is coplanar with point v i and point v i+1 , the Gaussian curvature is 0; after obtaining the Gaussian curvature, the photosynthetic photon flux density is taken as the x-axis, the temperature is taken as the y-axis, and the Gaussian curvature is taken as the z-axis for visualization, and the Gaussian curvature response surface is obtained.

[0025] In one embodiment, the step 4 introduces an incremental instantiation of the Gaussian curvature response surface in the photosynthetic photon flux density, to obtain the temperature-Gaussian curvature response curve under different light intensities, the input of the temperature-Gaussian curvature response curve is temperature, and the output is the curvature value; then the input and the output are normalized to eliminate the order of magnitude difference, and any point on the temperature-Gaussian curvature response curve is set as M i , and the u-string value thereof is calculated.

[0026] In one embodiment, two adjacent points M i and M j are generated by symmetric equidistant translation along the coordinate axis relative to the point M k , the two adjacent points maintain the Euclidean distance u with the point M i , and satisfy the following geometric constraints:

[0027]

[0028] In the formula, u is the topological distance, and 0<u<1;

[0029] The cosine value related to the included angle of the forearm vector in the support domain [M j , M k ] of M i is calculated to measure the size of the u-string value, and the u-string value of the point M i is calculated as follows:

[0030]

[0031] Wherein, s i = sign [(x i -x k )(y j -y k )-(x j -x k )(y i -y k )] is used to determine the positive and negative signs of the u-string value, (x i , y i ), (x j , y j ), (x k , y k ) represent the coordinates of the points M i , M j , M k .

[0032] In one embodiment, a random restart hill climbing method is used to obtain the upper and lower boundary points of the suitable temperature control target, and the obtained points are corresponded to the response surface of the photosynthetic rate prediction model, so that the corresponding temperature control interval is obtained, and the formula for obtaining the upper and lower boundary points is as follows:

[0033]

[0034] where c i and c j are the u-chord values of the upper and lower boundary points of the Gaussian response curve, respectively, and are any points on the left and right sides of c i , respectively, and are any points on the left and right sides of c j , respectively.

[0035] Compared with the prior art, the present application is based on curvature theory and aims to determine the optimal temperature regulation range for greenhouse peppers at different growth stages. First, a four-factor nested experiment is developed to study the effects of different temperatures, CO2 concentrations, and photosynthetic photon flux densities on the photosynthetic rate (Pn) of peppers at different growth stages. Subsequently, a genetic algorithm-optimized back propagation neural network is used to construct a photosynthetic rate prediction model, which exhibits high accuracy and reliability, with R 2 2 of 0.9812 and a MSE of 1.35 μmol·m -2 ·s -1 . Then, the prediction model is discretized and applied to the Gaussian curvature response surface calculation of Pn. Finally, the u-chord length algorithm and random restart hill climbing method are used to precisely define the boundaries of the temperature regulation range. Practical tests show that the average dry weight of the test group of peppers is 96.83% higher than that of the non-operated regulation group and 243.65% higher than that of the fixed threshold group. This result fully demonstrates that the method not only promotes the growth of peppers but also exhibits good regulation tolerance. The temperature regulation strategy of the present application provides key technical support and reliable decision-making basis for precision environmental management of greenhouse crops in facility agriculture. BRIEF DESCRIPTION OF DRAWINGS

[0036] Figure 1 is a schematic diagram of the Pn measurement process of the present application.

[0037] Figure 2 is a flowchart of the GA-optimized BPNN of the present application.

[0038] Figure 3 is a flowchart of the Gaussian curvature calculation of the present application.

[0039] Figure 4 is a schematic diagram of the U-chord curvature calculation of the present application.

[0040] Figure 5 is a simulated greenhouse temperature and PPFD variation curve, where (a) is the temperature variation curve and (b) is the PPFD variation curve.

[0041] Figure 6 is the light response curve of the present application, wherein (a) is CO2 concentration 300 μmol·mol -1 , (b) is CO2 concentration 900 μmol·mol -1 .

[0042] Figure 7 is the photosynthetic response surface of the present application, wherein (a) is CO2 concentration 600 μmol·mol -1 , (b) is CO2 concentration 900 μmol·mol -1 .

[0043] Figure 8 is the Gaussian curvature response surface and its thermogram of the present application.

[0044] Figure 9 is the result of the greenhouse temperature regulation range of the present application.

[0045] Figure 10 is the result of the biomass comparison and significance analysis of pepper under different regulation strategies of the present application, wherein (a) is plant height, (b) is stem diameter, (c) is fresh weight, (d) is dry weight, (e) is canopy leaf area, and (f) is seedling vigor index.

[0046] Figure 11 is the comparison of the number of days from the two-leaf-one-leaf stage to flowering and fruiting of pepper under three regulation strategies of the present application, wherein (a) is average daily flowering, (b) is average daily fruiting, (c) is average flower number, (d) is average fruit number, (e) is average fresh weight of fruit, and (f) is average dry weight of fruit. DETAILED DESCRIPTION

[0047] The embodiments of the present application will be described in detail below with reference to the accompanying drawings and examples.

[0048] In facility agriculture, extreme temperatures can cause irreversible damage to crops, so temperature regulation has become a key component of greenhouse environmental management. Dynamic optimization of temperature range is a core strategy to improve production stability. Therefore, determining the optimal temperature range is crucial for maximizing greenhouse production efficiency.

[0049] To solve this problem, the present application takes pepper as a sample, provides a curvature method-based optimal temperature regulation interval determination method for greenhouse pepper in the whole growth stage, and the method is based on the Gaussian curvature and u string length algorithm. The method takes the growth stage, CO2 concentration, temperature and PPFD as input variables, adopts a genetic algorithm-optimized back propagation neural network (GA-BPNN) to construct a Pn prediction model, and generates a corresponding Pn response surface. Then, the Gaussian curvature of the Pn response surface is calculated, the Gaussian curvature response surface is constructed, and the change trend of the Pn response surface is comprehensively reflected. Finally, the Gaussian curvature response surface is discretized into a temperature-Gaussian curvature (T-G) response curve, the u string value is taken as a measurement, and the random restart hill climbing method is used to determine the critical boundary of the response curve. The analysis process determines the optimal environmental regulation target range by systematically identifying these boundary points. The method comprehensively considers the interaction of various environmental factors, studies the suitable temperature interval of pepper in multiple growth periods, and provides a reliable basis for temperature regulation of facility agriculture.

[0050] In the embodiment of the present application, "Zhenyan" pepper is selected as a sample, and a nested test is carried out in the Agricultural Internet of Things Key Laboratory of the Ministry of Agriculture of Northwest A&F University from August 16 to October 4, 2023. The photosynthetic rate of pepper in different growth stages under different environmental conditions is measured, and a data set is constructed. The test sample is germinated and grown in a 72-hole seed tray, and then transplanted into a 20-cm-diameter and 16-cm-high seedling cup when it grows to 2 leaves and 1 heart. It is further cultured in an RGL-P500D-CO2 climate chamber (China Hefei Dasikate). The incubator is set to 12 hours of daylight, 25℃, 12 hours of night, and 20℃. The humidity is 50% throughout the day, and the CO2 concentration is 400 μmol·mol -1 . After the seedlings are recovered for one week, the Li-6800 portable plant photosynthesis tester (LI-COR, USA) is used to measure the Pn under the influence of various environmental factors in the whole growth period of pepper. When measuring, the pepper leaves are clamped in the leaf chamber of the Li-6800, as shown in Figure 1 . The present application measures the light intensity by photosynthetic photon flux density, and the data of the sample in the seedling stage, flowering stage, fruiting stage, fruiting stage, and fruiting stage are collected. Five temperature gradients, 10 photosynthetic photon flux density gradients, and 4 CO2 concentration gradients are set, and a total of 1000 groups of data (5x5x10x4=1000) are collected, as shown in Table 1.

[0051] A multi-dimensional environmental physiological response dataset containing 1000 data points (5x5x10x4=1000) was generated by comprehensive control of temperature, light intensity (PPFD) and CO2 concentration at the same batch of plants at consecutive key growth stages (seedling stage, flowering stage, pre-fruit stage, mid-fruit stage and post-fruit stage), as shown in Table 1. The continuous cross-cycle nested combination experiment of environmental factors improves the data acquisition efficiency and effectively reproduces the complex dynamic environmental interaction scenarios that crops may encounter throughout their life cycle. Therefore, it directly supports the engineering requirements of greenhouse environmental regulation.

[0052] Table 1 Environmental variable settings

[0053] Environmental factors Variable values Temperature (°C) 20,24,28,32,36 PPFD (pmol m -2 ·s -1 )]]> 0,50,100,200,300,600,900,1200,1500,1800 CO2 concentration (pmol-mol -1 )]]> 300,600,900,1200 CO2 concentration (pmol-mol -1 )]]> 300,600,900,1200

[0054] Step 2, model construction.

[0055] With growth stage, CO2 concentration, temperature and photosynthetic photon flux density as input vector, photosynthetic rate as output vector, the dataset is constructed, as shown in formula (1).

[0056] {(X1,Y1),(X2,Y2)…,(X n ,Y n )}(1)

[0057] Wherein, and represent the CO2 concentration, period, temperature and photosynthetic photon flux density of the i-th sample; Y i =[y i ], represents the photosynthetic rate measurement of the i-th sample.

[0058] In order to avoid significant deviation caused by different units and orders of magnitude between growth stage, CO2 concentration, temperature and photosynthetic photon flux density and photosynthetic rate, improve the convergence speed of the model and improve the performance of the model, the present application adopts a normalized data preprocessing method to convert these environmental variables and photosynthetic rate to a unified scale, mapping to the interval [0.2, 0.8], as shown in formula (2). Finally, the training set and the test set are randomly divided in the ratio of 7:3, the training set is used to train the photosynthetic rate model, and the test set is used to evaluate the generalization ability of the model.

[0059]

[0060] Wherein, x i ' is the normalized data, x i is the measured data under the i-th environmental condition, is the minimum value of the measured data, is the maximum value of the measured data.

[0061] Back propagation neural network (BPNN) can refine its weights and thresholds through back propagation mechanism, so as to realize regression prediction of data set. Since BPNN structure is intuitive and can effectively capture complex effects, the present application adopts genetic algorithm optimized back propagation neural network to construct photosynthetic rate prediction model. In order to comprehensively evaluate the effect of BPNN in Pn prediction, support vector regression (SVR) and polynomial regression (PR) are also compared and analyzed, which are two commonly used methods in Pn prediction.

[0062] BPNN has a three-layer feedforward structure, including input layer (4 nodes, corresponding to growth stage, CO2 concentration, temperature and photosynthetic photon flux density, four input variables), single hidden layer (10 nodes) and output layer (1 node, output photosynthetic rate prediction value). The back propagation method using mean square error (MSE) as loss function is used to train the model. The learning rate is set to 0.01, and the training is continued for 1000 times or until convergence. The specific calculation process is shown in formula (3):

[0063]

[0064] Where f(X) is the photosynthetic rate prediction value, w ij is the weight between the hidden layer and the input layer, w jk is the weight between the hidden layer and the output layer, b j is the threshold value between the hidden layer and the input layer, b k is the threshold value between the hidden layer and the output layer, k is the number of input variables, n is the data volume, h(·) is the Sigmoid activation function, which compresses the output to the interval [0,1], g(·) is the ReLU transfer function, is the tth feature under the ith environmental condition, t=1,2,3,4, representing growth stage, CO2 concentration, temperature and photosynthetic photon flux density, respectively.

[0065] SVR and PR are used as comparative models of BPNN to evaluate the performance of the system. SVR uses kernel function to handle nonlinear relationship, which converts low-dimensional nonlinear problem into high-dimensional linear space. This model uses radial basis function (RBF) kernel for Pn prediction, and optimizes hyperparameters through grid search method. PR uses four-variable quartic polynomial regression to achieve. This model simplifies complexity by taking the highest order term of temperature as the main influence, while combining the interaction of temperature factors.

[0066] In order to enhance the robustness of the model and improve the convergence efficiency, genetic algorithm can be used for parameter setting. The parameters to be optimized include the initial weights and thresholds of BPNN, the penalty parameter c and kernel function parameter g of SVR, and the coefficients a i of PR. Taking the optimization process of BPNN as an example, its flow chart is as followsFigure 2 The genetic algorithm defines the population size S = 100, the genetic generation G = 50, the crossover probability P x = 0.5, the mutation probability P c = 0.2, and the mutation gene number M = 3 of each chromosome.

[0067] To measure the performance and training effect of the model, the root mean square error (RMSE), the maximum absolute error (MAE), and the determination coefficient (R 2 ) and other evaluation indexes are used to evaluate the performance of the model, as shown in equations (4)-(6):

[0068]

[0069] MAE = max{|Y i -f(X i )|}, i = 1, 2,..., n (5)

[0070]

[0071] Where n is the number of data, and f(·) is the predicted value of the photosynthetic rate model.

[0072] Since photosynthesis plays an important role in crop biomass accumulation, excessive high or low temperature will affect crop yield. In order to determine the optimal growth temperature of peppers at different growth stages, the Gaussian curvature model is used to calculate the curvature change of the corresponding Pn response surface. This method significantly enhances the fault tolerance and stability of environmental dynamic regulation, and can more deeply identify the robust trend of Pn response to environmental changes, so as to determine the optimal growth temperature range of crops.

[0073] The Pn response surface is constructed with PPFD and temperature as x and y coordinates respectively. These surfaces integrate the effects of temperature, PPFD, CO2 concentration, growth stage, etc. on Pn, reveal the variation law of Pn, and provide guidance for determining the appropriate temperature target range. Gaussian curvature is mainly used to describe the curvature degree or shape feature of a point on the surface. The higher the value, the greater the curvature of the surface, and the lower the value, the flatter the surface. Therefore, the present application uses Gaussian curvature to explore the curvature change of the Pn response surface, laying a foundation for determining the target temperature range.

[0074] Step 3, construct the Gaussian curvature response surface.

[0075] By discretizing the Pn response surface into discrete points, the Gaussian curvature of each point is calculated to determine the Gaussian curvature of the Pn response surface. According to the design range of the experiment, the temperature discrete range is 20-36℃, the distance walking length is set to 0.1℃, and the discrete set is T = {t i}, t i=20+0.1i, i=0,1,2,…,160. Since the light compensation point of pepper is generally between 50-100 μmol·m -2 ·s -1 Therefore, the dispersion range of PPFD is set to 100-1800 μmol·m -2 ·s -1 , with a step size of 10 μmol·m -2 ·s -1 , the discrete set is P={P j}, where P j =100+10j, j=0, 1, 2, …, 170. Through this discretization process, a total of 27,531 discrete points are generated for each Pn response surface.

[0076] The three-dimensional data of photosynthetic photon flux density, temperature and photosynthetic rate are normalized to resolve the significant amplitude differences between the parameters. This preprocessing improves the stability of the curvature calculation. Then, the Gaussian curvature at each point is calculated to construct the Gaussian curvature response surface. In order to perform the calculation analysis, any discrete point of the photosynthetic rate response surface is taken as point P. There are four adjacent base points around point P. Corresponding to the position of its adjacent surface, the geometric configuration approximates point P as infinitesimally close to the base surface. Point P and its four surrounding base points form adjacent triangles, such as Figure 3 The Gaussian curvature K at point P is finally determined by angle summation and triangular area calculation using equations (7)-(9).

[0077]

[0078] Where N is the number of triangles formed by point P and its adjacent base points, and A is the sum of the areas of all triangles. Figure 3 In the middle, point P and point v i and dot v i+1 The triangle formed by θ i represents the vertex angle of the i-th triangle. The side length of the triangle is l i 、l i+1 and k i , s=(l i +l i+1 +k i ) / 2 represents the semi-perimeter of the triangle intersecting point P; if the local surface area around point P is flat, that is, point P and point v i and dot v i+1 If the Gaussian curvature is positive, the discrete surface resembles an ellipsoid. Conversely, if the Gaussian curvature is negative, the discrete surface has a saddle shape.

[0079] After obtaining the Gaussian curvature, the Gaussian curvature response surface is obtained by taking the photosynthetic photon flux density as the x-axis, the temperature as the y-axis, and the Gaussian curvature as the z-axis.

[0080] Step 4, the Gaussian curvature response surface is discretized into a temperature-Gaussian curvature response curve, and the critical boundary of the response curve is determined by using the random restart hill climbing method with the u chord value as the metric, to obtain the optimal temperature regulation interval.

[0081] The Gaussian curvature response surface can reveal the response law of Pn to temperature changes of pepper at different growth stages, CO2 concentrations and PPFD conditions, which provides an important basis for determining the target regulation boundary. In order to accurately and intuitively identify the temperature regulation boundary, first, the Gaussian curvature response surface is instantiated with 50 μmol·m -2 ·s -1 as the step size on the x-axis of the Gaussian curvature response surface. Specifically, the curve relationship between temperature and Gaussian curvature, i.e., the temperature-Gaussian curvature (T-G) response curve, is obtained under different light intensities (for example, under the conditions of PPFD of 50 μmol·m -2 ·s -1 , 100 μmol·m -2 ·s -1 , 150 μmol·m -2 ·s -1 , etc.). The independent variable of the T-G response curve is temperature, and the dependent variable is the curvature value. Subsequently, the independent variable and the dependent variable are normalized using equation (2) to eliminate the magnitude difference. According to the change law of the response curve, a point M i is randomly selected on the T-G response curve. Based on point M i , two adjacent points M j and M k are generated by symmetric translation along the coordinate axes relative to point M i . The translation operation is performed in the two-dimensional coordinate system of the T-G response curve, ensuring that the Euclidean distance between the two points M j and M k and point M i is u (topological distance, 0<u<1), and satisfies the geometric constraints of equation (10) and equation (11), as shown in Figure 4 .

[0082] ||M j M i ||=u (10)

[0083] ||M i M k ||=u (11)

[0084] In the formula, u is the topological distance, 0<u<1.

[0085] Find M j and M k Two points, you can get M i Support areas[M j ,M k ] Calculate M i Support areas[M j ,M k ]The cosine value of the angle between the front and rear arm vectors is used to measure the size of the u chord value. i The u chord value of is shown in formula (12):

[0086]

[0087] Among them, s i =sign[(x i -x k )(y j -y k )-(x j -x k )(y i -y k )] is used to determine the positive and negative signs of the u chord value, (x i ,y i ),(x j ,y j ),(x k ,y k ) represent points M i , M j , M k 's coordinates.

[0088] The u-chord value of the TG response curve quantifies the rate or degree of change in Gaussian curvature with temperature. It can be used as an important metric for determining boundary targets. Large u-chord values ​​indicate that Gaussian curvature (i.e., the sensitivity of photosynthetic rate to temperature) changes more dramatically around that temperature. Based on the u-chord value, reasonable upper and lower boundaries are selected on either side of the TG response curve's peak as the target temperature range. Figure 9 (a) shows the specific trend of the TG response curve. Due to the complex physiological phenomena of peppers, the curvature trends on both sides of the peak are different. On the left side of the peak, the TG response curve shows a trend of first decreasing and then increasing. This reflects that in the process of temperature change, the intensity of Pn's response to temperature change increases significantly at a certain moment. Therefore, this drastic change point is found as the upper boundary, corresponding to the u chord value. Its corresponding change is also first weakened and then increased. Therefore, within this range, the minimum u chord target value c is found. iThe upper boundary of the temperature range can be obtained. On the right side of the peak, the T-G response curve gradually tends to be stable, indicating that the growth trend of Pn with temperature is slowly weakened or even starts to decline, therefore, the critical point of Pn change with temperature before the significant weakening is found as the lower boundary of temperature. The change rule of u string value is first small, then large, and then small, and the turning point of u string value from large to small is found, which can determine the lower boundary of temperature. Therefore, according to this rule, the random restart hill climbing method is adopted, and the upper and lower boundary points of the suitable temperature regulation target are obtained according to the two strategies of formula (13) and (14). Finally, the obtained points are corresponded to the response surface of the photosynthetic rate prediction model, and the corresponding temperature regulation interval can be obtained.

[0089]

[0090]

[0091] wherein c i and c j are the u string values of the upper and lower boundary points of the Gaussian response curve, and are any points on the left and right sides of c i , and are any points on the left and right sides of c j .

[0092] In the embodiment of the application, from August 18 to October 18, 2024, the samples of 'Zhenyan' pepper were cultured to the flowering stage, and the long-term culture verification test was completed, and the influence of the optimal temperature range determined by the application on the growth of the pepper was evaluated. In the five growth stages of the pepper, the seedling stage is the key period of root development, leaf formation and nutrient accumulation. It is very sensitive to temperature, which directly affects their yield and quality. After entering the flowering stage, the growth focus of the pepper changes from vegetative growth to reproductive growth, resulting in a slow expansion of branches and leaves. Therefore, morphological and physiological phenotypes, including plant height, stem diameter, dry weight, fresh weight, leaf area and seeding vigor (formula 15) are used to evaluate the growth of the pepper seedling stage. In addition, the number of flowers and fruits, the number of days required for flowering and fruiting, and the dry weight and fresh weight of the fruits are used as evaluation indexes for the flowering and fruiting stage.

[0093] The experiment was divided into one experimental group and two control groups. Using the summer greenhouse data of Northwest A&F University Jingyang Vegetable Experimentation Demonstration Station, an artificial climate chamber was used to simulate environmental changes. This simulated environment was designed based on real greenhouse data, ensuring that the scenarios in the validation test are highly representative. Therefore, its results can directly provide information for the development of actual greenhouse control strategies. The control phase is from 8:00 to 20:00. Temperature and PPFD are adjusted every 2 hours, while the average CO2 concentration in the first 15 minutes is used as the model input. This control frequency evaluates the model's ability to capture greenhouse environmental changes while maintaining stability, which represents a key method to address the inherent uncertainty in real-world environmental monitoring data. Figure 5 The environmental control for the first 5 days is shown. The experimental group adjusts the temperature in the direction of the optimal temperature according to the temperature range determined by the present application. One control group is cultured in a climate chamber to simulate the greenhouse temperature fluctuations as shown in Figure 5 , and the other control group is cultured at a fixed temperature of 25℃. All other environmental variables of the three groups are consistent with the environmental variables recorded at the time of data collection. The above indicators are analyzed using Minitab 2022 software.

[0094]

[0095] In the formula, S is the stem diameter, H is the plant height, RDW is the root dry weight, SDW is the stem dry weight, and TDW is the total dry weight.

[0096] According to the experimental data, the effects of temperature and environmental factors on Pn at different stages are explored and presented in the form of light response curves, as shown in Figure 6 . Comparative analysis shows that there are significant differences in light response curves at different growth stages, and temperature is a key factor affecting photosynthetic characteristics. At low CO2 concentration (300 μmol·mol -1 ), the flowering stage shows more aggregated light response curves than other flowering stages, and the Pn difference between adjacent temperature gradients (4℃) remains at 1 μmol·m -2 ·s -1 Below, showing strong thermal adaptability (20-32℃). While other growth periods, especially the seedling stage, have a 2.5 times higher curve dispersion than the flowering stage. When the temperature exceeds 32℃, PPFD≥800 μmol·m -2 ·s -1 , the seedling stage Pn decreases by an average of 83.24% under 28℃ conditions, indicating increased environmental sensitivity. When the CO2 concentration increases (900 μmol·mol -1 ), the warming effect on photosynthetic capacity is enhanced, with Pn increasing by 10-16 μmol·m -2 ·s -1, the light response curve distribution was expanded. This CO2-temperature interaction significantly reduced the high temperature / light co-suppression, especially in the seedling stage and fruit middle stage. At the same time, the sensitivity of plants to temperature was enhanced, effectively buffering the heat fluctuations. These dynamic coordinated thermal responses provide a key framework for precise modeling of photosynthetic mechanisms under climate change scenarios.

[0097] Pn in different growth stages was predicted using BPNN, SVR, and PR and their GA-optimized versions. The prediction performance of these models was based on RMSE, MAE, and R 2 The evaluation was conducted, and the comprehensive results are shown in Table 2. Among these methods, GA-BPNN showed excellent performance in all indicators, with the highest R 2 value (0.9812) and the lowest RMSE and MAE (1.35 μmol·m -2 ·s -1 and 0.89 μmol·m -2 ·s -1 , respectively). GA-SVR was second, and PR was the worst. In addition, the fitting slope of the GA-BPNN model was 0.9752, indicating that its prediction results were closer to the ideal scenario (1:1 line). These results show that BPNN has robust nonlinear mapping capabilities, and after optimization by genetic algorithm, it can achieve higher prediction accuracy. This provides a solid theoretical basis for future research to determine the optimal growth temperature range.

[0098] Table 2 Comparison of evaluation results of test set for algorithms

[0099]

[0100] The above results show that compared with other models, the BPNN model shows better performance in prediction accuracy and applicability, and is more suitable for subsequent research on temperature adjustment range. In order to intuitively show the prediction performance of GA-BPNN, the Pn response surface is shown in Figure 7 Figure 7 Fig. 4 (a) and (b) are the prediction results of BPNN for Pn in the flowering stage and fruit pre-stage when the CO2 concentration is 600 μmol·mol -1 and the CO2 concentration is 900 μmol·mol -1 , respectively.

[0101] ​Although Pn trends vary at different stages and environmental conditions, it is worth noting that Pn changes are often positively correlated with temperature changes. This indicates that within the highest temperature set in the invention (36℃), the photosynthetic enzyme does not denature or inactivate. However, as the temperature rises, the rate of increase in Pn gradually slows down, indicating that high temperatures inhibit the photosynthesis of peppers. Therefore, in-depth exploration of the potential variation of Pn of peppers at different growth stages is of great significance for reasonable planning of the appropriate temperature regulation range.

[0102] In order to comprehensively study the global response law of pepper Pn to temperature changes, the Gaussian curvature of the Pn response surface was calculated, and a Gaussian curvature response surface model was constructed. The purpose is to quantify and reflect the curvature characteristics of the Pn response surface, as shown in Figure 8 The model can intuitively show the Gaussian response characteristics of the pepper Pn response surface under different conditions. Figure 8 For CO2 concentration of 600 μmol·mol -1 , flowering stage and CO2 concentration of 900 μmol·mol -1 , pre-fruiting stage Pn response surface corresponding to the Gaussian curvature response surface and its thermal map.

[0103] Figure 8 In (a), the Gaussian response surface under different environments and periods is presented, and both models show high similarity in shape in response to temperature. Figure 8 In (b), the quantitative analysis of the curvature parameter found that the interaction of temperature and PPFD significantly affected the shape of the response surface. When PPFD is lower than about 800 μmol·m -2 ·s -1 , the Gaussian curvature is negative (K<0), showing a two-phase feature of first rising and then falling with temperature, and finally tending to be stable, and the response surface forms a concave surface. Combined with Figure 8 observation data, it can be known that within this PPFD interval, within the temperature range of 20-36℃, the increase of temperature only leads to a weak increase of Pn, and the increase amplitude does not exceed 5 μmol·m -2 ·s -1 . However, the absolute value of the curvature can reach more than 0.20, which indicates that at this time, temperature is not the main factor limiting the development of Pn, and the interaction of light intensity and temperature on Pn presents an antagonistic effect, making the pepper have a wide range of temperature adaptation, which is consistent with the meaning of negative curvature.

[0104] On the contrary, when PPFD is higher than 800 μmol·m -2 ·s -1At this time, the Gaussian curvature gradually changes to positive values (K > 0), showing an opposite trend with temperature, and the response surface forms a convex. In this case, Pn grows faster with temperature than under low light intensity conditions. When reaching light saturation, temperature becomes the main limiting factor for Pn. The high-temperature inhibition effect makes the light and temperature constraint directions consistent, driving Pn growth in a convex acceleration mode. This change enhances the temperature sensitivity of peppers, narrowing their adaptive temperature range. In a greenhouse environment, high light intensity often occurs simultaneously with high temperature, making the fixed threshold adjustment method neither accurate nor cost-effective. Balancing temperature adjustment within the optimal range by slightly sacrificing Pn becomes a key priority for greenhouse management.

[0105] Notably, inflection points in the temperature dynamics of Pn were observed in the ranges of 20-24 °C and 32-36 °C, which may be related to the phase transition temperature of photosynthetic enzyme kinetics. These inflection points elucidate the adaptive mechanisms of pepper photosynthesis under different temperature conditions, which are of great significance for understanding the temperature sensitivity of pepper growth. Therefore, determining the temperature adjustment boundary between 20-24 °C and 32-36 °C is crucial for optimizing the temperature management of pepper cultivation greenhouses. This finding lays a solid scientific foundation for advancing precision agriculture practices in controlled environments.

[0106] However, the Gaussian curvature response surface has a complex geometry, and points on the surface can exhibit different curvatures and directions, making it difficult and resource-intensive to identify and calculate critical boundaries. At the same time, while the Gaussian curvature method can effectively capture the global changes in Pn, noise present in the complex surface can interfere with the selection of feature points. This interference complicates the accurate delineation of temperature range boundaries, further affecting the precision of temperature adjustment. Therefore, the introduction of the u-string algorithm, which has advantages in feature point selection, is particularly important. This integration reduces the computational dimension, enabling more accurate determination of the optimal temperature range for pepper growth, ultimately achieving more efficient and precise cultivation management.

[0107] To better reflect the prominent features of the Gaussian curvature response surface and facilitate the exploration of critical boundaries, the Gaussian curvature response surface was converted into multiple T-G response curves. Then, according to the actual planting conditions, the point with the maximum rate of change to the left of the peak and the point with the minimum rate of change to the right were selected as the upper and lower boundaries of the optimal temperature adjustment range, respectively. The peak value is the optimal adjustment temperature value. This method not only improves the precision of sunlight greenhouse temperature management but also benefits the growth and development of crops. On this basis, the u-string value was used as a measure to determine the upper and lower boundary points, as shown in Figure 9 .

[0108] As shown in Figure 9 , under the flowering stage of CO2 concentration of 600 μmol·mol -1 and CO2 concentration of 900 μmol·mol-1 The optimal temperature ranges under 7 different PPFD levels in the early fruiting stage. Figure 9 (a) shows the temperature range of the TG response curve, which shows that the optimal growth temperature range for crops varies at different growth stages and under different PPFD conditions. Generally speaking, the optimal growth temperature range for peppers narrows with increasing PPFD. This is because strong light induces photoinhibition of the pepper photosynthetic system, while the combined inhibitory effect of high temperature limits normal growth. Therefore, under strong light conditions, peppers show greater sensitivity to temperature changes, resulting in a narrower optimal growth temperature range. Conversely, under weaker PPFD conditions, peppers show greater adaptability to temperature changes, resulting in a wider optimal temperature range. Figure 9 (b) shows the two-dimensional mapping of the temperature range corresponding to the Pn response surface. The two green lines represent the upper and lower boundaries of the adjustment range, and the black dashed line with an asterisk in the middle represents the optimal temperature. Figure 9 As shown in (b), the optimum temperature range during the pre-fruiting stage is significantly narrower than during the flowering stage. This difference occurs because the transition from vegetative to reproductive growth involves more complex nutrient allocation, making pepper plants more sensitive to temperature requirements. Excessively high temperatures can cause trunk elongation and fruit drop. This also confirms the need to manage pepper greenhouse temperatures according to different stages. More detailed results are shown in Table 3. As can be seen, as PPFD increases from 300 μmol·m -2 ·s -1 Increased to 900 μmol·m -2 ·s -1 The optimum temperature range of each growth environment and growth stage is generally shrinking. Figure 9 In addition, Table 3 clearly reflects the temperature requirements of pepper plants: slightly lower temperatures in the seedling stage are beneficial for vegetative growth, while higher temperatures in the mid-fruiting stage are beneficial for fruit development. In summary, the determined temperature range effectively avoids the high temperature inhibition zone and provides a reasonable temperature management strategy for pepper cultivation.

[0109] Table 3 Optimal temperature ranges under different environments and periods

[0110]

[0111] In order to verify the influence of different environments on the optimal temperature range of peppers, the method of the present invention was compared with the fixed threshold adjustment method. The results are shown in Table 4. Table 4 gives the CO2 concentration of 600 μmol·mol -1 Some results of Figure 5 It can be seen that PPFD in summer greenhouse environment generally does not exceed 1000 μmol·m -2 ·s -1, therefore Table 4 only shows the PPFD results within this range. Compared with the fixed threshold method, the average Pn growth rate of the method proposed in the present invention was 10.68%, indicating that temperature regulation can significantly improve photosynthetic efficiency. However, the seedling stage showed contradictory characteristics: although its Pn increment was the smallest (only 3.34%, significantly lower than other growth stages), its absolute Pn value was the highest among all growth stages. This phenomenon indicates that during seedling development, the response threshold of photosynthesis to temperature changes is close to saturation, at which time slight thermal fluctuations may seriously affect physiological performance. This behavior may be due to the fact that seedlings prioritize organogenesis rather than stress response mechanisms, resulting in reduced heat tolerance, resilience and adaptive temperature range. These findings are consistent with the data in Table 3.

[0112] Table 4 Comparison of the optimal temperature range determined by the present invention and the fixed threshold method

[0113]

[0114] Furthermore, analysis of the average temperature ranges for each pepper growth stage (Table 4) reveals that, compared with the acclimated temperature of approximately 26.3°C during the seedling stage, the temperature requirements for other growth stages increase significantly, particularly reaching approximately 28.1°C during the mid-fruiting phase. This indicates that pepper's temperature adaptability varies throughout its growth. This confirms the feasibility and necessity of adjusting greenhouse temperatures according to physiological stages.

[0115] The present invention has confirmed the significant advantages of the dynamic range control strategy in pepper production through system verification experiments. Figure 10 As shown, compared with the control group of simulated summer greenhouse conditions (No-operation) and fixed threshold regulation, the regulation strategy proposed in the present invention has stronger adaptability and better comprehensive benefits. Specifically, the average fresh weight of peppers in the experimental group was 63.37% higher than that of the no-operation regulation group and 16.13% higher than that of the fixed threshold regulation group. The average dry weight was 78.19% higher than that of the no-operation regulation group and 20.25% higher than that of the fixed threshold regulation group. In terms of stem thickness and seedling vitality index, the optimal amplitude regulation method was significantly better than the condition without regulation strategy. Although there was no significant difference compared with fixed threshold regulation, the overall effect was better. As Figure 10As shown in (e), under the optimal control range, pepper plants achieved the largest canopy leaf area, increasing by 34.83% and 23.22% compared to no manipulation and fixed threshold control, respectively. These results further validate the multidimensional synergistic enhancement effect of this method, which is more beneficial to pepper growth. Combined with the seedling data shown in Table 4, although the increase in Pn was modest, it significantly promoted pepper plant growth. This phenomenon can be attributed to the comprehensive consideration of other environmental factors and growth stages when selecting the optimal temperature range, thereby improving overall photosynthetic efficiency and achieving cumulative growth benefits. These findings collectively confirm the rationality and applicability of the proposed temperature management strategy in guiding agricultural production practices.

[0116] The flowering and fruiting of peppers cultivated with three strategies are as follows Figure 11 Compared with the other two strategies, the flowering and fruiting performance of peppers grown within the optimal temperature range were significantly improved. Figure 11 As can be seen in (a) and (b), compared with the fixed threshold and no operation strategies, the average flowering days of this group were advanced by 4-7 days. The results showed that the two treatments shortened the average fruiting period by 8.29 days and 14.15 days respectively. In terms of the number of flowers and fruits, this group has a clear advantage over the other two groups. Figure 11 As shown in (c) and (d). Comparative analysis showed that the fixed threshold strategy extended the plant's growth period, while the non-manipulation environment disrupted the growth dynamics of peppers, leading to an imbalance of excessive vegetative growth at the expense of reproductive development. This imbalance led to delayed flowering and fruiting, as well as a high incidence of flower and fruit abscission. In addition, Figure 11 Panels (e)-(f) further demonstrate the superior fruit quality achieved by regulating the optimal temperature range. The average fresh fruit weight of the experimental group increased by 225.57% compared to the unmanipulated group and by 91.53% compared to the fixed threshold group. Dry weight increased by 243.65% and 96.84% relative to these two baselines, respectively. These findings confirm that maintaining an optimal temperature range not only accelerates pepper phenology (shortening the growth cycle) but also improves reproductive outcomes by increasing flower / fruit yield and fruit biomass.

Claims

1. A method for determining the optimal temperature control interval of greenhouse peppers during the entire growth stage based on a curvature method, characterized in that: The steps include: Step 1: measuring the photosynthetic rate of peppers at different growth stages under different environmental conditions through nested experiments to construct a data set; the environmental conditions include temperature, CO2 concentration, and photosynthetic photon flux density; Step 2: Using the growth stage, CO2 concentration, temperature, and photosynthetic photon flux density as input vectors, a back propagation neural network optimized by genetic algorithm is used to construct a photosynthetic rate prediction model and generate the corresponding photosynthetic rate response surface; Step 3, discretizing the photosynthetic rate response surface into discrete points, calculating the Gaussian curvature at each point, and constructing a Gaussian curvature response surface; Step 4: discretize the Gaussian curvature response surface into a temperature-Gaussian curvature response curve, use the u chord value as a measure, and use the random restart hill climbing method to determine the critical boundary of the response curve to obtain the optimal temperature control range.

2. The method for determining the optimal temperature control interval of the whole growth stage of greenhouse pepper based on the curvature method according to claim 1 is characterized in that: The growth stages include the seedling stage, the flowering stage, the pre-fruiting stage, the mid-fruiting stage and the late-fruiting stage. The different environmental conditions are to set a number of temperature gradients, a number of photosynthetic photon flux density gradients and a number of CO2 concentration gradients.

3. The method for determining the optimal temperature control interval of the whole growth stage of greenhouse pepper based on the curvature method according to claim 1 is characterized in that: The data obtained through nested experiments are normalized using the following formula: Among them, x i ′ is the normalized data, x i is the measured data under the i-th environmental condition, is the minimum value of the measured data, is the maximum value of the measured data.

4. The method for determining the optimal temperature control interval of the whole growth stage of greenhouse pepper based on the curvature method according to claim 1 is characterized in that: The back propagation neural network has a three-layer feedforward structure, including an input layer, a single hidden layer, and an output layer. The model is trained using the back propagation method with mean square error as the loss function. The calculation process is as follows: Where f(X) is the predicted value of photosynthetic rate, w ij is the weight between the hidden layer and the input layer, w jk is the weight between the hidden layer and the output layer, b j is the threshold between the hidden layer and the input layer, b k is the threshold between the hidden layer and the output layer, k is the number of input variables, n is the amount of data, h(·) is the Sigmoid activation function, which compresses the output to the interval [0,1], g(·) is the ReLU transfer function, is the t-th feature under the i-th environmental condition, t = 1, 2, 3, 4, representing the growth stage, CO2 concentration, temperature and photosynthetic photon flux density, respectively.

5. The method for determining the optimal temperature control interval of the whole growth stage of greenhouse pepper based on the curvature method according to claim 4 is characterized in that: The input layer has 4 nodes, corresponding to the four input variables of growth stage, CO2 concentration, temperature and photosynthetic photon flux density, the hidden layer has 10 nodes, and the output layer has 1 node, which outputs the predicted value of photosynthetic rate.

6. The method for determining the optimal temperature control interval of greenhouse peppers in the whole growth stage based on the curvature method according to claim 1 is characterized in that: The step 2 is to construct a photosynthetic rate response surface using photosynthetic photon flux density and temperature as x-coordinates and y-coordinates; the step 3 is to normalize the photosynthetic photon flux density, temperature and photosynthetic rate to resolve significant amplitude differences between the parameters.

7. The method for determining the optimal temperature control interval for the entire growth stage of greenhouse peppers based on the curvature method according to claim 1 or 6, characterized in that: In step 3, any discrete point of the photosynthetic rate response surface is taken as point P. There are four adjacent base points around point P. Corresponding to the position of its adjacent surface, the geometric configuration approximates point P as being infinitesimally close to the base surface. Point P and its four surrounding base points form an adjacent triangle, and the formula is as follows: Where N is the number of triangles formed by point P and its adjacent base points, and A is the sum of the areas of all triangles. i and dot v i+1 The triangle formed by θ i Represents the vertex angle of the i-th triangle, and the side length of the triangle is l i 、l i+1 and k i , s=(l i +l i+1 +k i ) / 2 represents the semiperimeter of the triangle intersecting point P; If the local surface area around point P is flat, that is, point P and point v i and dot v i+1 If they are coplanar, the Gaussian curvature is 0. After obtaining the Gaussian curvature, the photosynthetic photon flux density is taken as the x-axis, the temperature is taken as the y-axis, and the Gaussian curvature is taken as the z-axis for visualization to obtain the Gaussian curvature response surface.

8. The method for determining the optimal temperature control interval of greenhouse peppers in the whole growth stage based on the curvature method according to claim 1 is characterized in that: In step 4, an incremental instantiation of the Gaussian curvature response surface is introduced into the photosynthetic photon flux density to obtain a temperature-Gaussian curvature response curve under different light intensities, wherein the input of the temperature-Gaussian curvature response curve is the temperature and the output is the curvature value; the input and output are then normalized to eliminate the magnitude difference, and any point on the temperature-Gaussian curvature response curve is set as M i , calculate its u chord value.

9. The method for determining the optimal temperature control interval for the entire growth stage of greenhouse peppers based on the curvature method according to claim 8, characterized in that: By relative to point M i Generate two adjacent points M by symmetrically equidistant translation along the coordinate axis j and M k , the two adjacent points and M i Maintain the Euclidean distance u and satisfy the following geometric constraints: ||M j M i ||=u ||M i M k ||=u Where u is the topological distance, 0 <u<1; Calculate M i Support areas[M j ,M k ]The cosine value of the angle between the front and rear arm vectors is used to measure the size of the u chord value. At this time, point M i The u-chord value of is calculated as follows: Among them, s i =sign[(x i -x k )(y j -y k )-(x j -x k )(y i -y k )] is used to determine the positive and negative signs of the u chord value, (x i ,y i ),(x j ,y j ),(x k ,y k ) represent points M i , M j , M k 's coordinates.

10. The method for determining the optimal temperature control interval of greenhouse peppers in the entire growth stage based on the curvature method according to claim 8, characterized in that: The random restart hill climbing method is used to obtain the upper and lower boundary points of the suitable temperature control target. The obtained points are mapped to the response surface of the photosynthetic rate prediction model to obtain the corresponding temperature control range. The formula for obtaining the upper and lower boundary points is as follows: Among them, c i and c j are the u chord values ​​of the upper and lower boundary points of the Gaussian response curve, and c i Any point on the left or right side of and c j Any point on the left or right side of .

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