Bioelectromagnetics. 1995;16(3):197-206.
Measurement and Analysis of Static Magnetic Fields That Block Action Potentials in Cultured Neurons
Cavopol AV, Wamil AW, Holcomb RR, McLean MJ
INTRODUCTION
Electrically stimulated action potentials of adult mouse sensory neurons in cell culture were blocked to a large extent when the neuron was positioned in a static magnetic field of -11 mT intensity produced by an array of four permanent magnets of alternating polarity [McLean et al., 1991,1995]. In the original experiments, the magnitude of this biological effect depended on positioning of the neuron under study in the field (changes in vertical distance of cell from the array), on array parameters (e.g., action potential blockade decreased with increasing distance between magnets in the array), and on the number and polarity of the magnets in the array.
To facilitate description of the field properties that determined the biological effect, we have developed a mathematical formalism that closely simulates the experimental field measurements. Computerassisted modeling facilitates a comprehensive algebraic characterization of the field properties. Here we compare field characteristics of different arrays to identify possible correlations between field structure and the corresponding AP blockade.
MATERIALS AND METHODS
The methods regarding cell culture, action potential recording and data analysis have been published previously McLean et al., 1991, 19951.
Magnetic Field Measurements
The magnetic field produced by arrays of cylindrical neodymium magnets (radius a = 7 mm and height h = 5 mm, referred to as macroarruy) was measured with a gaussmeter (see Appendix) model 4048 (F.W. Bell, Newton, MA). The probe was attached to an x,y,z, mechanical micromanipulator, and the fields were scanned from the surface of the magnet outward along the z axis at 1 mm increments and sideways in the other two dimensions at a constant height, z = 6 mm, for all three field components BX, By, and BZ (Fig. 1). The three scanning directions intersected at a site corresponding to the cell position in the field.
With reference to the center of the array, designated (x,y,z) = (O,O,O)t,h e cell position throughout the experiments was in the range(x,y,z) = (0 f 1 mm, 4.5 f 1 mm, 6 I!I 1 mm). Positioning of the magnets and the experimental apparatus have been described in detail elsewhere [McLean et al., 1991, 19941. For some experiments, magnets of radii a = 0.2 mm and height h = 1 mm were used to construct microarrays with an overall diameter of -0.8 mm. Some dimensions relevant to the microarray experiments are given for future reference:
The neuronal diameter was aCELL= 40- 60 p and the physical microarray area diameter was a? ARRAY. -800 pm. The neuron under study was positioned at a height h = 1.5 sfr 0.2 mm above the array. Positioning in the horizontal plane was difficult due to the small size of the array. Positioning variability contributed to a wide range of effects on action potential firing (see Discussion and Appendix). The background magnetic field and its variation over the cell culture dish were below 1 gauss.
Experimental Error
A detailed error analysis is presented in the Appendix. Measurement errors were caused mainly by vernier reading error, by imperfect superposition of the coordinate systems of the magnet and the micromanipulator arm holding the probe, and by gaussmeter reading error, which was considered small compared to the other two. Another experimental limitation comes from the field averaging performed by the gaussmeter due to the size of the measurement window of the Hall probe.
Computer Model
A computer model of array magnetic fields supplements our measurements, allows three-dimensional graphic display of the fields and makes quantitative analysis of field characteristics possible. To preserve relative algebraic simplicity, we assume that. the field of a cylindrical magnet can be described by the magnetic field of a current loop (magnetic dipole) [Landau and Lifschitz, 19691.
The field generated by a fourmagnet array is obtained by superposition of the four individual dipole fields. The resultant total field is characterized by three parameters: a = radius of the magnet, 2c = distance between adjacent magnet centers, and m = a scaling factor.
RESULTS
Theoretical Fit Plots of experimental data and simulations based on these data were within the experimental measurement error for most of the domain. For a parameter setting that corresponds to the physical magnet dimensions (a = c = 7 mm), the field amplitude has been set to match the peak value of the experimentally measured component BXZ.
This sets the value for the scaling parameter m (m = 43) and completely specifies the theoretical field, which is then compared to the remaining components of the experimental field. As an example, the fit between the measured and computed values of the x component of the field is shown in Figure 2.
The computer model closely reproduces all the experimental field measurements (additional data available upon request), especially in the data collection range (0 & 1 mm, 4.5 ?I 1 mm, 6 &- 1 mm) where field variations are linear. Computed values for the y and z components fit the measured data equally well. This validates use of the computer model as an analytic tool to describe the global characteristics of the field.