Two-Dimensional Brain Microtubule Structures Behave as Memristive Devices

Abstract

Microtubules (MTs) are cytoskeletal structures that play a central role in a variety of cell functions including cell division and cargo transfer. MTs are also nonlinear electrical transmission lines that produce and conduct electrical oscillations elicited by changes in either electric field and/or ionic gradients. The oscillatory behavior of MTs requires a voltage-sensitive gating mechanism to enable the electrodiffusional ionic movement through the MT wall. Here we explored the electrical response of non-oscillating rat brain MT sheets to square voltage steps. To ascertain the nature of the possible gating mechanism, the electrical response of non-oscillating rat brain MT sheets (2D arrays of MTs) to square pulses was analyzed under voltage-clamping conditions. A complex voltage-dependent nonlinear charge movement was observed, which represented the summation of two events. The first contribution was a small, saturating, voltage-dependent capacitance with a maximum charge displacement in the range of 4 fC/μm². A second, major contribution was a non-saturating voltage-dependent charge transfer, consistent with the properties of a multistep memristive device. The memristive capabilities of MTs could drive oscillatory behavior, and enable voltage-driven neuromorphic circuits and architectures within neurons.

Figure 1

Experimental setup and electrical response of voltage-clamped MT sheet. (a) (Top) Schematics of electrical setup to obtain electrical signals from giga-sealed MT sheets. (Bottom) Rat brain MT sheet with attached patch pipette. (b) Top tracing (Left), typical oscillations from a patch-clamped MT sheet in symmetrical 140 mM KCl. Bottom tracing is a typical record of a silent MT sheet under similar experimental conditions. Only “silent” (i.e. non-oscillatory) MT sheets were utilized in the present study. (Right) Power spectra from both tracings show the fundamental frequency (29 Hz) only in the active MT sheet. (c) Representative electrical response to a 5 msec-voltage clamping protocol between ±100 mV, starting from a holding potential of zero mV (as shown in Top). Resting time in between voltage steps was 45 msec. (d) MT patch conductance obtained by the current-to-voltage relationship for the tracings shown in (c). Values (filled circles) were obtained at the time point indicated by R_ss. The solid line indicates the ohmic response as shown from the best fitting to a straight line. Similar values were obtained from 20 msec voltage steps (open triangles). The R_ss for this patch was 2.63 ± 0.04 GΩ.

Figure 2

Time-dependence of the relaxation response. (a) Current transient obtained in response to the transition from the end of the voltage step to zero mV. Time constants τ were obtained at 0.37 I_peak (maximal deflection current) from both the ON and OFF transitions at the onset and ending times of the voltage step, respectively. Inset. The relaxation response was multi-exponential, as indicated in the Linear-Log plot. Blue line indicates the mono-exponential function fitting. (b) Plot showing voltage dependence of τ vs. V_h for the onset (ON, filled circles), and ending (OFF, open circles) of the pulse.

Figure 3

Calculation of displaced charge during current transients. (a) Blown up current transients obtained in response to a voltage step. Peak (I_peak) and steady state (I_ss) current values are shown for both the onset (ON), and ending (OFF) values. Total charge transferred, Q_total, (shaded area) for each region was obtained by numerical integration of the total area under the curve. Similar calculations were carried out for each current transient at the onset and ending of the pulse. (b) Q_total obtained from experiments conducted in symmetrical KCl (140 mM), was plotted vs. V_h, and shown for the entire ±100 mV voltage range. Experimental points (filled circles) are the mean ± SE, of n = 9 experiments. The solid line depicts best fitting to Eq. 7. Q_total showed no saturation and a nonlinear voltage response with inflection around zero mV. (c) Plot of the patch capacitance C_MT (Eq. 8), as a function of V_h. The experimental data (filled circles, n = 9), were fitted to Eq. 9 (Red Line). (d) Plot of the patch capacitance C_MT (Eq. 8), as a function of V_h after addition of 200 mM K-gluconate to the bathing solution following conditions as before. The experimental data (filled circles, n = 6), were fitted to Eq. 9. Solid lines represent best fittings for symmetrical KCl (Red) and asymmetrical (Green) conditions. (e) Data in (b) were corrected by subtraction of linear response showing the displacement charge vs. V_h and fitted to a Boltzmann function (Eq. 7 without the linear term), as shown by the Red Line under symmetrical conditions and Green Line after addition of K-gluconate. (f) The slope of the Q_OFF vs. Q_ON plot indicated the capacitive nature of the translocated charge.

Figure 4

Current-to-voltage relationship of non-oscillating MT sheets. (a) Temporal correlation between train of voltage steps between ±100 mV (V_h, Bottom), and elicited current spikes (I, Top). Blown up region of a single pair is shown below. (b) The plot shows the current-to-voltage response from data in (a). Note that each vertical line is actually a spike as shown in the Inset. The solid red line connects the peak values of the spikes, emphasizing the nonlinear, saturating response of I_peak vs. V_h. Please note that the portion of linear solid line closed to the abscissa represents the linear, ohmic response of I_ss vs. V_h. Data representative of n = 9 experiments. (c) (Left) 3D-plot of an electrical response from a non oscillating MT sheet after ±100 mV train of pulses both holding potential and current response were plotted vs time. (Right) The expanded tracing shows the electrical response to −100 mV and −95 mV to indicate the zero current-zero voltage instances as indicated by the black arrows. Red arrows help in following the time response.

Figure 5

Memristive properties of MT sheet current spikes. (a) (Left) Schematics of theoretical total charge Q (Top) and total φ (Bottom) obtained by integration of either current spikes (Top tracing) or voltage steps (Bottom tracing), respectively. Please note positive pulses are in red and negative pulses in black. (Right) Graph showing representative φ vs. Q plot for voltage steps between ±100 mV. (b) Temporal response of current transients for voltage steps between ±5 mV (Left) and ±100 mV (Right), and best fitted theoretical lines (black and red solid lines, respectively) following Eq. 1. Note that negative currents are shown as positive deflections.