Diffusion Influenced Adsorption Kinetics

When the kinetics of adsorption is influenced by the diffusive flow of solutes, the solute concentration at the surface is influenced by the surface coverage of solutes, which is given by the Langmuir–Hinshelwood adsorption equation. The diffusion equation with the boundary condition given by the Langmuir–Hinshelwood adsorption equation leads to the nonlinear integro-differential equation for the surface coverage. In this paper, we solved the nonlinear integro-differential equation using the Grünwald–Letnikov formula developed to solve fractional kinetics. Guided by the numerical results, analytical expressions for the upper and lower bounds of the exact numerical results were obtained. The upper and lower bounds were close to the exact numerical results in the diffusion- and reaction-controlled limits, respectively. We examined the validity of the two simple analytical expressions obtained in the diffusion-controlled limit. The results were generalized to include the effect of dispersive diffusion. We also investigated the effect of molecular rearrangement of anisotropic molecules on surface coverage

Relationship between response latency and amplitude of the DR- and SR-evoked potentials.

<p>(A) Plots of the DR-evoked potentials against latency and amplitude. The 57 responses observed from 179 recording sites in monkey SO are shown. (B) Same as A, but for monkey TA. The 50 responses of 168 recording sites are shown. (C, D) Same as (A, B), but for the SR-evoked potentials. The 163 responses observed from 179 recording sites in monkey SO (C) and 87 responses observed from 168 recording sites in monkey TA (D) are shown.</p

Characteristics of the SR-evoked potentials in area 3a.

<p>(A) Percent responses evoked by DR and SR stimulations in area 3a. (B) Percent responses evoked by DR and SR stimulations in area 3b/1. (C) Amplitude of the DR- and SR-evoked potentials in area 3a. (D) Latency of the DR- and SR-evoked potentials in area 3a. Error bars in A–D indicated S.E. Asterisk in A–D indicates statistical significance at <i>P</i> < 0.05 using two-sample <i>t</i>-test.</p

Schematic drawing of experimental setting and surface map of the recording sites.

<p>(A) Stimulation of deep radial (DR) and superficial radial (SR) nerves. Three nerve cuffs were implanted: one on the radial nerve trunk (R) at the left forearm, one on the DR, representing primarily muscle afferent input, and one on the SR, representing primarily input from the skin. The DR and SR cuffs were used for electrical stimulation, and the R cuff was used for recording incoming volleys. The nerves were stimulated with biphasic constant-current pulses, 100 μs/phase, at twice the threshold (2T). The electrical stimulation-evoked field potential was recorded from the forearm region at the posterior bank of the CS of the right hemisphere. (B) Cortical surface map of the recording sites in each monkey. The electrode was inserted 8–15 mm at the anterior–posterior level. Gray lines indicate the approximate location of the CS on the cortical surface. Recording sites of the SR- or DR-evoked potentials are indicated by filled circles. Open circles indicate electrode insertions in which intracortical microstimulations were applied and no SEPs were recorded. Body parts activated at the lowest current of the microstimulation are indicated by capital letters. Values indicate the lowest microstimulation current (μA) evoking the movement. “n” indicates no effect up to 200 μA. A: anterior, P: posterior, M: medial, L: lateral.</p

Examples of DR- and SR-evoked potential distribution in area 3a.

<p>(A) Amplitude plots of the DR-evoked potentials at the anterior-posterior 10-mm level in monkey SO where the largest DR-evoked potential was observed in this animal. The circle size indicates the amplitude. A bar indicates no significant activation. (B) Latency plots of the DR-evoked potential indicated by colors. (C, D) same as (A, B), but for the SR-evoked potentials. (E–H) Same as A-D but for monkey TA. The data in A11 were from where the largest DR-evoked potential was observed in this monkey. The scale bar indicates 5 mm, separated by black and gray every 1 mm.</p

Experimental setup.

<p><b><i>A</i></b>, Dorsal view of the spine and dorsal root ganglions (DRGs) in the cervical segments of a monkey. The cervical (C2) through thoracic (Th3) vertebrae are shown. Microelectrode arrays were implanted in the left C7 and C8 DRGs. <b><i>B</i></b><b>,</b> Marker placements on the left arm and hand.</p

Total number of units used in the model and number of units selected by SLiR using single units of putative cutaneous units only.

<p>The values in the columns are total number of units used in the model (Total) and number of putative cutaneous units selected by the SLiR (Selected cutaneous) in encoding of joint angle (Angle), velocity (Vel.), and acceleration (Accel.) of the elbow, wrist, and D5 MCP joints. Data are expressed as the mean ± standard deviation (n = 10).</p

Representative examples of DR- and SR-evoked potentials.

<p>(A) An example of the DR- and SR-evoked potentials recorded at the bottom of the posterior part of CS. Averaged traces of 60 trials are shown at the time aligned with DR and SR stimulations. An asterisk indicates a detected response compared with the baseline activity. Blue dotted lines indicate one standard deviation from the baseline activity. (B) Same as (A) except for recording at the superficial part of the posterior part of CS. (C) Reconstructed trace of the electrode insertion in which the SEPs in A and B were recorded. (D) An example of the thionine-stained coronal section in monkey SO. White arrows indicate the approximate locations of the borders of areas 4, 3a, 3b, and 1. Colored arrows indicate the lesion marks. The scale bars in C and D indicate 5 mm, separated by black and gray every 1 mm. (E) Amplitude plots of the DR- and SR-evoked potentials. The potentials were recorded every 1 mm from the surface to the bottom of the posterior part of CS as shown in C. (F) Latency plots of the DR- and SR-evoked potentials. In C, E, and F, characters A and B indicate the recording depth of the SEPs represented in A and B, respectively.</p

Overlaying plots of the DR- and SR-evoked potentials for all recording sites.

<p>(A) Amplitude of the DR-evoked potentials for all recording sites in monkey SO. The circle size indicates the amplitude. A bar at each recording site indicates no significant response. (B) Latency plots of the DR-evoked potentials for all recording sites in monkey SO. The latency is indicated by colors. The scale bar indicates 5 mm, separated by black and gray every 1 mm. (C, D) Same as (A, B), but for the SR-evoked potentials. (E–H) Same as A–D, but for monkey TA. All SEPs from multiple anterior-posterior level were overplayed into one typical histology section in each monkey.</p

Response property of units selected by the SLiR.

<p><b><i>A</i></b><i>,</i> Peri-stimulus time histograms (bin width 50 ms) for two units recorded during joint movements and skin brushing. Elbow joint angle and positions of the paint brush are shown in the bottom row, in which black lines and dark gray areas represent the mean and standard deviation, respectively. A light gray area represents duration in which the paint brush touched to skin surface. <b><i>B</i></b><i>,</i> Number of all recorded units (original) and those selected units by the SLiR (selected) belonging to the respective class in Monkey 1.</p