To minimize these ambiguities and to develop a more robust approach for non-invasive in-situ root imaging, we aim to develop iCSD inversion code that does not rely on prior assumptions on root architecture and function and use rhizotron experiments to validate the iCSD approach.The phrase “inversion of Current Source Density” was introduced by Łęski et al. to describe the 2D imaging of current sources associated with the brain neural activation. Similar inversion methodologies have been developed for the interpretation of the self potential data, where the distribution of naturally occurring currents is investigated . With regard to active methodologies, Binley et al. developed an analogous approach for detecting pollutant leakage from environmental confinement barriers. Although there are physical and numerical intrinsic differences between application of the iCSD to detect brain neuronal activity and current pathways in roots, we decided to adopt the term iCSD as the general physical imaging of current source density remains valid. With iCSD, we indicate the coupling of ERT and MALM through the proposed numerical inversion procedure for the imaging of the current source density,hydroponic net pots and its correlation with root architecture. We introduce the necessary aspects regarding the ERT and MALM methods in this section. However, we direct the interested readers to more in-depth discussion about the ERT method , and to Schlumberger and Parasnis with regard to the MALM method.
In the following discussion we use ρmed to represent the 2D or 3D distribution of the electrical resistivity in the growing medium . CSD represents the 2D, or 3D, distribution of the Current Source Density within the same medium. In the case of roots, the CSD is controlled by the current conduction behavior of the roots, specifically by the leakage pattern of the root system . Both ERT and MALM are active methods. In these methods the current is forced through the medium by applying a potential difference between two current electrodes. In ERT, both current electrodes are positioned in the investigated medium, while for MALM the positive current pole is installed in the plant stem,similar to BIA . The potential field resulting from the current injection depends on CSD, resistivity of the medium , and boundary conditions. The boundary conditions are known a priori and their impact on the potential field can be properly modeled. In ERT, the current sources correspond to the electrodes used to inject current, allowing us to invert for ρmed. Then, the iCSD accounts for the obtained ρmed and explicitly inverts the MALM data to obtain current source distribution.The rhizotrons used in this study were designed to enable the concurrent direct visualization of the roots and electrical measurements. Rhizotron dimensions were 52 cm × 53 cm × 2 cm , see Fig. 2. Figure 2a shows the rhizotron setup with 64 silver/silver chloride electrodes located on the back viewing surface.
The viewing surfaces were covered with opaque material to stop the light from affecting the development of the roots. The back viewing surface was removable, allowing homogeneous soil packing for the plant experiments and convenient access to the electrodes. Besides the top opening, the rhizotrons were waterproof to enable hydroponic experiments and controlled evapotranspiration conditions during the soil experiments and plant growth. All the experiments were performed in a growth chamber equipped with automatic growth lights and controlled temperature and humidity. The temperature varied with a day/night temperature regime of 25/20 °C. The humidity ranged from 45 to 60%. For both ERT and MALM methods, the electrical potential field is characterized by a set of potential differences measured between pairs of electrodes. It is important to properly arrange the electrodes on the rhizotron viewing surface and design a suitable acquisition sequence to obtain a good sensitivity coverage of the investigated system . This is particularly true for the iCSD, as both ERT and MALM acquisitions affect its result. The 64 electrodes were arranged in a 8 by 8 grid on the back viewing surface of the rhizotron, leaving the front surface clear for the observation . For the ERT, the designed arrangement of the electrodes offers a good compromise between a high coverage on the central part of the rhizotron, which encompasses the root zone, and a sufficient coverage on the rhizotron sides to avoid an excessive ERT inversion smoothness. For the MALM, the arrangement of the electrodes is highly sensitive to the position of the investigated current sources. Because of their central positions, the electrodes are closer to the expected sources of current and thus in the region of maximum potential gradient. Hence, this electrode configuration maximizes the changes in both magnitude and sign of the measured ΔV associated with a change in the CSD distribution.
The electrode diameter was 1.5 mm. The penetration of the electrodes into the rhizotron was 4 mm ± 1 mm. To evaluate the possible distorting effects of the densely populated electrodes on the potential field distribution, a test was performed with low conductivity water . The test showed no resistivity anomalies, which may be caused by the presence of the electrodes . Therefore, while rhizotron setups with electrodes only on the sides were successfully adopted , we found that the current setup represents a better solution for iCSD experiments .For the ERT acquisition over the 2D grid of electrodes, we chose a dipole-dipole skip 2 configuration. For each skip 2-couple of injection electrodes the remaining skip-2 couples of electrodes were used as potential dipoles . The associated complete set of reciprocals was also acquired, the resulting acquisition sequence contained 3904 data points . Following the ERT data acquisition, the MALM data acquisition required little setup adjustments and time. As the two current electrodes are fixed, the use of a multichannel resistivity meter significantly reduced the acquisition time and, consequently, supported the acquisition of more robust data sets. Electrode 1 was used to inject the current into the plant stem, while electrode 64 was used as a return electrode in the growing medium . The remaining 62 electrodes were used to map the resulting potential field. A sequence with 204 ΔVs was used. Considering the grid in Fig. 2a, the sequence included the vertical, horizontal, and diagonal ΔVs between adjacent electrodes. While 61 ΔVs would provide all the independent differences,blueberry grow pot the 204 ΔV sequence was preferred because of its redundancy and consequent lower sensitivity to acquisition errors. The acquisition time remained relatively short as the multichannel instrument was optimized with fixed current electrodes that allowed 8 ΔVs to be measured at once.
The iCSD inversion that we developed was based on the physical principles of a bounded system in which linearity and charge conservation were applied to decompose the investigated CSD distribution into the sum of point current sources. This provided a discrete representation of the root system portions where the current leaks from the roots into the surrounding medium. Because of the linearity of the problem, the collective potential field from multiple current sources is the linear combination of their individual potential fields. As such, the measured ΔV can be viewed as and decomposed into the sum of multiple ΔVs from a set of possible current sources. These possible current sources are namedViRTual electrodes . As purely numerical electrodes, they are simulated by mesh nodes representing possible current sources, but with no direct correlation with the real electrodes used during data acquisition. Basically, the VRTe were distributed to represent a grid over which the true CSD distribution is discretized. In order to account for any possible CSD, a 2D grid of 306 VRTe was arranged to cover the entire rhizotron . The charge conservation law implies that the sum of the current fractions associated with the VRTe has to be equal to the overall injected current, which is provided by the resistivity meter. If we normalize the injected current to be equal to 1, the sum of the VRTe weights has to be 1 as well. Briefly, for Ohm’s law, normalizing the current to 1 is equivalent to calculating the resistance, R, from ΔV. Then, the use of R simplifies the presentation of the numerical problem. Once the VRTe nodes are added to the ERT-based ρmed structure, the potential field associated with each of the VRTes is simulated with BERT. From these simulated potential fields, the same sequence of 204 R is extracted, each corresponding to a single VRTe. Each extracted sequence contains the resistances that would be measured in the laboratory if all the current sources were concentrated at the VRTe point .Synthetic numerical and laboratory experimental tests were performed in order to evaluate the capabilities of the setup and inversion routine to couple the ERT and MALM approaches for the iCSD. In the numerical tests both the true source response and VRTe responses were calculated with BERT. Figure 3 shows an explanatory numerical test with inversion of a point source, and the associated Pareto front that was used to select the optimum regularization strength. As this first experiment was performed to specifically test the inversion routine, a homogeneous ρmed was used in order to avoid influence from the baseline resistivity distribution complexity. For the second experiment, the laboratory tests were conducted. Because of the ρmed heterogeneity of any experimental system, these laboratory tests need to include the ERT inversion, and the use of the obtained ρmed as input in the iCSD.
The true current sources were obtained using insulated metallic wires inserted into the rhizotron . The insulating plastic cover was removed at the tips of the metallic wires to obtain the desired current sources. Six experimental tests were performed using different numbers and positions of these current sources. The rhizotron was filled with tap water and left to equilibrate to achieve steady state conditions of water temperature and salinity, thus minimizing ρmed heterogeneity and changes during the experiment. Changes in ρmed during the ERT and MALM acquisition periods would make the ERT-based ρmed less accurate and compromise the iCSD. To make sure ρmed was stable, a second ERT was performed after the MALM acquisition and compared with the initial measurement. The conductivity of the solution was also measured in several locations of the rhizotron with a conductivity meter to validate theρmed obtained from the ERT inversion. This setup allowed the acquisition of good quality data sets since less than 5% of the data were discharged during the data processing. Because of the controlled laboratory conditions, the ρmed obtained with the ERT was stable and consistent with the direct conductivity measurements. The quality of the ERT inversion was also confirmed by comparing the model responses with the acquired data . Similarly, the acquired iCSD data were plotted against the resistances calculated with the CSD distribution obtained from the iCSD. The tests also allowed a more informed definition of the VRTe grid. For our setup, a spacing of 3 cm provided a good compromise between resolution, stability, and duration of the iCSD routine. The 3-cm spacing also agrees with the ERT resolution, which would not support a higher iCSD resolution. Successive numerical tests were based on the 8- source laboratory tests shown in Fig. 4. These tests aimed to 1) link laboratory and numerical tests to evaluate the influence of the numerical iCSD routine and laboratory setup on the overall iCSD stability and resolution; 2) account for a more complex CSD, given by the 8 wire-tip sources that were used to simulate distal current pathways; and 3) account for possible ρmed heterogeneity. To address goals 1 and 2, the position of the 8 sources was replicated in the numerical tests and a test with homogeneous ρmed was included to simulate the water resistivity of the laboratory tests. To address goal 3, heterogeneous ρmed were tested.In order to account for the heterogeneous ρmed the following modeling steps were carried out. First, a true ρmed was assigned to the mesh cells of the rhizotron ERT model. We included homogeneous, linear, and quadratic resistivity profiles in the y direction, see Fig. 5. Second, the ERT acquisition was simulated with the ERT laboratory sequence and 3% of Gaussian error, in line with reciprocal and stacking errors observed in the laboratory data sets. Third, the forwarded ERT data sets were inverted following the exact laboratory procedure.