Neuronal networks connected by electrical synapses, also referred to as gap junctions, are present throughout the entire central nervous system. Many instances of gap-junctional coupling are formed between dendritic arbours of individual cells, and these dendro-dendritic gap junctions are known to play an important role in mediating various brain rhythms in both normal and pathological states. The dynamics of such neuronal networks modelled by passive or quasi-active (resonant) membranes can be described by the Green’s function which provides the fundamental input-output relationships of the entire network. One of the methods for calculating this response function is the so-called ‘sum-over-trips’ framework which enables the construction of the Green’s function for an arbitrary network as a convergent infinite series solution. Here we propose an alternative and computationally efficient approach for constructing the Green’s functions on dendro-dendritic gap junction-coupled neuronal networks which avoids any infinite terms in the solutions. Instead, the Green’s function is constructed from the solution of a system of linear algebraic equations. We apply this new method to a number of systems including a simple single cell model and two-cell neuronal networks. We also demonstrate that the application of this novel approach allows one to reduce a model with complex dendritic formations to an equivalent model with a much simpler morphological structure. Neuronal cells have a distinctive structure which differentiates them from any other cell types. The most extended parts of many neurons are dendrites, and their morphological complexity has fascinated scientists since the exemplary work of Ramón y Cajal . OrganisThe dendritic membrane of various types of neurons is known to be equipped with voltage-gated ion channels, nonuniformly distributed throughout dendritic arbours and often demonstrating nonlinear dynamics. Many models of neuronal cells with retention of complex dendritic formations are built by combining the linear (passive) properties of dendrites together with nonlinear (active) dynamics of ion channels. At the level of a single cell or at the network level, such models are restricted to being solved only by numerical methods, based on a compartmental approach . Althougkinetics .Here we focus on a network of neuronal cells with purely passive or resonant membrane dynamics coupled by dendro-dendritic electrical synapses, also known as gap junctions. Gap junctions are mechanical and electrically conductive links between adjacent neuronal cells that permit direct electrical connections between them. Having been first discovered at the giant motor synapses of the crayfish in the late 1950s, gap junctions are now known to be expressed in the majority of cell types in the brain , 13. UsiIn Sect. We consider a network of neuronal cells. Each cell consists of an arbitrary structure of a dendritic morphology and a lumped soma, and cells in the network are connected by gap junctions  on the left and right from a gap junction   2) for ementclass2pt{minimmentclass2pt{minimmentclasspt{minimaX on branch i in response to a Dirac delta pulse applied to the location Y on branch j at time The whole network in Fig. i:k of the network by its own characteristic function i and ends at the point j. The trips coefficients Initiate Branching node: entclass1pt{minimaSomatic node: entclass1pt{minimaGJ node: entclass1pt{minimaTerminal: We refer the reader to [Our method is based on and developed from the sum-over-trips approach for calculating the Green’s function on a network of electrically coupled neuronal cells . Considementclass2pt{minimive (see ) that thy can be divided into two classes separated by the direction of the last part of the trip. Placing two points j as shown in Fig. y from the left . The function n to segment k and trip coefficients k. These coefficients are obtained from the sum-over-trips rules summarised in Fig. Place a pair of points entclass1pt{minimaj which includes point y,  can support such structures.v, w) takes the following formHere we consider a simple model of a dendrite with a lumped soma attached to it at see Fig. . We assuund from  and 18)\documentx and y are placed at the soma. This dependence is obtained as a solution of the implicit equationThe model can be easily modified for the case of a semi-infinite dendrite. Assuming rom Eqs.  and 25)\documentnt}Jy in  takes thHere we demonstrate how our method can be applied to a two-cell network of either identical or nonidentical cells coupled by a dendro-dendritic gap junction. In each case, we obtain the compact solutions for the Green’s functions, Eqs. –43) for for43) fN attached semi-infinite dendrites as shown in Fig. We start by considering a model of two identical cells, either of which consists of a soma and given by . The gapUsing our method, we can construct a linear system of algebraic equations for the functions see Fig. :36\documrom Eqs.  and 19)\documentIn Fig. hat Eqs.  and 41)\documentAssuming that two distal inputs m of Eq.  for the k, k, and Using our method, we can also construct analytical solutions for the Green’s functions for a network of two nonidentical cells. The response functions at the soma of Cell 2 take the formsn by Eq. .Using Eqs. –56), we, we56), Often it is difficult to measure experimentally locations and strengths of gap junctions in real neuronal networks. Knowledge of the inverse mapN dendritic branches, one of which is the primary dendrite with the tuft spanning from its end. Two cells are coupled in their tufts by dendro-dendritic gap junctions  asing Eqs.  and 88)89\documementclasspt{minimamentclasspt{minimak can receive a Dirac delta pulse at the location Using our method of local point matching, we can also prove that there exists an equivalent reduced model for the full tufted model with external inputs applied to the tufts (instead of the primary dendrites). We consider that any tuft dendrite It is possible to show (see “Appendix 5” for details) that the Green’s function of the full tufted model for a given input y away from the primary dendrite of each cell, we obtain in this special caseIn the case of multiple inputs applied to the tuft dendrites, the Green’s function for each cell can be found by summing individual Green’s functions for each input. Assuming that all tuft dendrites of both cells receive identical inputs located at a distance In this paper we have presented a novel method for calculating the Green’s functions for arbitrary neuronal networks with a passive or resonant cell membrane coupled by dendro-dendritic gap junctions. This method provides an alternative and complementary approach to the generalised sum-over-trip method . ImportaN dendritic segments, we need to construct and solve a linear system of 2N equations only once to find all unknown X can be placed on each dendritic segment before constructing a system of linear equations for X points except one on branch i in the solution for There are a number of natural extensions of the work in this paper. One is an application to more realistic network geometries with more than just two cells, given that a computational implementation of the method of local point matching can provide a fast realisation of the Green’s function for the whole network. Having a complex network of multiple cells with a graph structure consisting of


This paper describes how the FyrisSKZ web tool can be used to evaluate allocation scenarios using data from the Svärta River, an agricultural catchment located in central Sweden. Three scenarios are evaluated: a baseline, a uniform 6-m-wide buffer zone in each sub-catchment, and an allocation of areas of buffer zones to sub-catchments based on the average cost of reduction. The total P reduction increases by 30 % in the second scenario compared to the baseline scenario, and the average reduction per hectare increases by 90 % while total costs of the program fall by 32 %. In the third scenario, the average cost per unit of reduction (€163 kg P No eutrophication is one of the 16 environmental quality objectives adopted by the Swedish Parliament. In a recent study by the Swedish Environmental Protection Agency, it was identified as an objective that will not be achieved by the target year 2020 under the current set of policies , and in Sweden, each of the five Swedish Water Districts is responsible for ensuring good water status including non-eutrophic status. cies SNV . Eutrophcies SNV . The Swecies SNV . MeasureRiparian buffer zones are the only measure which has been used extensively in Sweden to reduce phosphorus losses from agricultural land. Buffer zones primarily lower phosphorus losses through reducing erosion of particulate P from fields. The effectiveness of a riparian buffer zone depends on the parameters which have an effect on surface runoff  and the phosphorus load to the zone. Implementation of riparian buffer zones in Sweden has been supported by payments to landowners from the EU Rural Development Program (RDP). These payments have been a uniform reimbursement per hectare for buffer zones from 6 to 20 m wide in eligible areas for a 5-year commitment. The reimbursement is compensation for the average loss of income for developing the zone and taking the land out of agricultural production, the opportunity cost of the land. This opportunity cost is related to the productivity of the land and varies to a large extent based on agronomic factors. The level of payments has led to uneven and low participation in the program SLU .The purpose of this study is to demonstrate how a low transaction cost tool can be used to evaluate and design cost efficient programs for implementation of buffer zones. The first section below describes the design of agri-environmental schemes with a focus on targeting to improve cost efficiency. This section includes a discussion of transaction costs and their impact on policy design and how models by reducing these types of costs can improve the total cost efficiency of programs. The following section describes a web tool developed for evaluating the cost efficiency of buffer zones on reducing phosphorus losses from agricultural land in Sweden, FyrisSKZ. The next section demonstrates how the FyrisSKZ web tool can be used for program evaluation by using the model to compare the cost efficiency of three program scenarios in the Svärta River catchment in Southern Sweden. The paper ends with discussion and conclusions based on the results from the application.agri-environmental expenditures should be more precisely targeted;there should be a higher rate of EU contribution for sub-measures with a higher environmental potential;there should be a clear distinction between simple and more demanding agri-environment sub-measures;and that the Member States should be more proactive in managing agri-environment payments. are designed to reduce transaction costs; this includes the use of tools which can provide access to series of harmonized databases. The models described in the following section represent this type of tool and serve to provide information about the cost efficiency of buffer zones targeted at the reduction of P losses from the agricultural landscape. The low transaction costs associated with this tool allow users to evaluate allocation scenarios at a high level of resolution and achieve low cost targeting.http://fyrisskz.slu.se) which allows users to choose one or several sub-catchments from a GIS interface and view the estimated reduction of a buffer zone on phosphorus losses to the watercourse from surrounding fields, the opportunity and maintenance costs for buffer zones, and the potential area of buffer zones in the chosen sub-catchment. These estimates are presented for five individual buffer zone widths to allow the user to study the influence of the selected width on the reduction effect, costs, and potential area. The web tool was developed under an assignment from the Swedish Water Authorities to provide user friendly access on the cost efficiency of buffer zones based on data from the FyrisCOST DSS. Both the FyrisSKZ and FyrisCOST models were developed by the SLU WaterHUB group at the Swedish University of Agricultural Sciences (http://www.slu.se/en/collaborative-centres-and-projects/slu-water-hub/).The FyrisSKZ model is a web tool which summarizes the cost efficiency of buffer zones along lakes, watercourses, and ditches for all 12 864 sub-catchment areas of Sweden. FyrisSKZ is a public domain web application  which includes the ICECREAMDB  on the agricultural block data. All water courses were assigned a 15-m width and converted to lines in the landscape using the “st_boundary” function in PostgreSOL/PostGIS. Water courses (including ditches) which had a length of less than 30 m along an agricultural block were excluded from the potential area in the sub-catchment. The potential impact area was assumed to be the area in those non-excluded agricultural blocks that was within 50 m of a watercourse, and a 60 m buffer from the watercourse was used to delineate this area.Figure For statistical purposes, Sweden is divided into eight production regions where agricultural land within each of the regions has similar production capabilities. The FyrisCOST model is used to calculate the transport of P from each sub-catchment based on the region the sub-catchment is located within and additional information for each particular sub-catchment: soil types, crop distribution, point sources, and mitigation measures. Transport of P is recalculated to eliminate the effect of existing buffer zones in the catchment, and this P-transport baseline is then imported into FyrisSKZ to use for calculating the effect of buffer zones in the sub-catchment. Where there is a potential for a buffer zone  with 7500 ha of this in crop production. There are two dominant soil types in the catchment: silty clay loam (80 %) and silty loam. The majority of the soil has a high soil P concentration. As described above, the FyrisCOST model uses a three-tier system for P classes, and of the 13 sub-catchments in the Svärta River catchment, there are 11 of these in the highest class, 1 in the middle, and 1 in the lowest class. Erosion sensitive agricultural land (0–50 m from watercourses) is also divided into three slope classes with a distribution in the catchment of 11 in the highest class (greatest slope) and 2 in the middle class.To illustrate how the FyrisSKZ tool can support policy analysis, three scenarios for allocations of riparian buffer zones in the Svärta River catchment were evaluated. The Svärta River catchment is located in central Sweden south of Stockholm and drains directly to the Baltic Sea. Of the total catchment land area of 345 kmThe baseline scenario (Scenario 1) in the study uses the actual distribution of buffer zones which received subsidies from the Swedish RDP in 2008 and estimates the effect and costs of buffer zones used for calculation of the HELCOM PLC5 data (10 m wide) using output from the FyrisSKZ tool. The PLC5 data used a 10 m buffer zone width for all of the buffer zones because input data were only available for the total area of buffer zones subsidized through the RDP in each sub-catchment and not for specific widths (from 6 to 20 m) that were eligible under program guidelines. Scenario 2 allocates a 6-m-wide buffer zone on all the potential area in each sub-catchment. Potential area is based on the requirements for buffer zones along watercourses as defined in the Swedish RDP guidelines and estimated using the FyrisSKZ tool. Scenario 3 allocates areas of buffer zones (widths) to sub-catchments based on the lowest cost of reduction to achieve a total P reduction similar to the baseline scenario (Scenario 1).−1 with an average reduction of 0.60 kg P for every hectare in the program. The total P reduction is increased by 30 % in Scenario 2 compared to the baseline scenario, and the average reduction per hectare increases by 90 % while the total costs of the program fall by 32 %. The increased reduction and the constant cost per hectare of the land in buffer zones lead to a 45 % fall in the average cost per unit of P reduced: from €390 to €207 kg P−1. In the third scenario, the average cost per unit of reduction (€163 kg P−1) is the lowest of the three scenarios: 58 % lower than the baseline and 21 % lower than Scenario 2. In addition, as a result of having the lowest amount of land in buffer zones (71.5 ha), this scenario also has the lowest total program costs: 56 % lower than the baseline and 35 % lower than Scenario 2.Table In Scenario 3, a program targeted at implementing 6 m buffer zones in all potential areas in six sub-catchments results in a similar reduction to the baseline scenario at a lower cost and at a lower cost than having 6 m zones in all the sub-catchments as under Scenario 2. In spite of the lowest average cost per unit of P reduction in Scenario 3, it is not clear that this is the most cost efficient of the three scenarios. Selecting the most cost efficient scenario depends on the goal set by program administrators. If the goal is to maximize total P reduction, then Scenario 2 would be the preferred scenario as this leads to a higher reduction compared to the other two scenarios. However, if the goal is to achieve the lowest cost per unit of reduction, then Scenario 3 is the preferred scenario. The targets for P reduction under the BSAP would tend to support the adoption of Scenario 2 as this provides the greatest total reduction.Using a uniform payment to landowners, as assumed under all three scenarios, may make it difficult to achieve voluntary adoption of 6-m-wide zones everywhere in the catchment where there is a potential for a buffer zone. Voluntary programs require sufficient payment and sufficient information to achieve expected uptake levels. While the second of these two requirements would increase the transaction costs for a targeted program, reducing the total cost of the program (Scenarios 2 and 3 compared to the baseline) could make this possible. For example, under the RDP program in 2008, the total costs for buffer zones in the Svärta River catchment were around €38 000 while under Scenario 3, these were less than €17 000. The €21 000 difference in the two program costs could be used to promote adoption in the six targeted sub-catchments. This could perhaps be in the form of targeted media campaigns, visits to landowners by agricultural advisors or some other type of informational outreach. The problem of payment sufficient to induce participation is in part addressed by using a land cost based on reported leasing costs at the 90th percentile. While using this land cost may result in economic rents for some landowners with marginally productive land, it may be adequate for including the majority of productive land in a program. An alternative would be to adopt a policy which allowed for individually negotiated land payments, but this could also significantly raise transaction costs and lead to lower cost efficiency.Finally, it is important to point out that the two alternative scenarios evaluated (Scenarios 2 and 3) are only a subset of the total number of possible scenarios. The types of programs evaluated could be extended to include other targets. For example, if the administrative target was to achieve the same P reduction as under the baseline scenario for the region covered by one of the Swedish Water Authorities, this would include a large number of catchment areas and may lead to some catchments with no reduction and other catchments with greater reductions than if the target was set for each individual catchment. There are also other scenarios that could be of interest based on other criteria. Regardless of the type of program evaluated, since the FyrisSKZ tool covers all the 12 864 sub-catchments in Sweden, it is possible to evaluate and compare any type of program at any scale above the level of a single sub-catchment.Tools similar to FyrisSKZ could be developed for evaluation and design of other agricultural mitigation programs. Current plans include developing tools based on the same platform for evaluating two types of measures: catch crops and structural liming. The possibility of not only evaluating single programs of measures but also combinations of measures would make it possible for local authorities to design cost efficient action programs for meeting the targets of the WFD and the BSAP. If similar types of models and sufficient data were available, the FyrisSKZ type of tool could be developed in other countries.