All of the qualitative observations made for the horizontal case hold true, albeit with higher values of the exponent due to the larger current speed. This value is beyond the range of the figure. Only strictly positive values of the propagation rate c are physically meaningful. Upon examining the values in tables 1 and 2 , we observe that the propagation rate is positive for all parameter values; therefore, no restriction is placed on the problem parameters on this account.
The dependence of the rate of temporal variation of the current thickness F 2 on model parameters is illustrated in table 3 for horizontal widening channels and squeezing fractures. The first row of table 3 summarizes the expressions for F 2 , while the second row lists the conditions under which the thickness decreases or remains constant with increasing time. Only this case is asymptotically meaningful, as for positive values of F 2 , the current eventually reaches the top of a given channel or fracture.
For squeezing fractures, the analysis is more complex as a second threshold b t becomes important, depending on the channel shape k. Physically, this behaviour occurs because in squeezing fractures the available volume shrinks along the flow direction; therefore, both the rate of growth of the current volume and the squeezing parameter must be limited for the current to become thinner over time. Dependence of the thickness time exponent F 2 on the model parameters for horizontal widening channels and squeezing fractures.
Finally, the exponent F 2 decreases with b for widening channels and increases for squeezing fractures, as larger values of b imply an increase or decrease of available channel volume for these two cases, respectively. The dependency of F 2 on model parameters for the inclined case follows a similar pattern: Therefore, the exponent F 2 is always negative, and the thickness decreases with time for any combination of model parameters.
Physically, this means that when the motion is driven by gravity, the current becomes more elongated with time irrespective of channel geometry and rheology. The same is true in horizontal widening channels, as the viscous resistances decrease along the flow direction, and a smaller free-surface gradient is needed to drive the flow. In horizontal squeezing fractures, the aspect ratio decreases with time only if the squeezing is modest, or for intermediate rates of squeezing and moderate influx rates.
The validity of the thin-current approximation can also be checked at finite times according to, e. The layout of the experimental apparatus is depicted in figure 6. The two devices were used for tests with Newtonian or shear-thinning fluids, respectively, for flow rates below and above 3.
Shear-thickening fluids were directly poured in the channel through funnels of different sizes, refilling the fluid manually to ensure a constant delivery rate. Layout of the experimental apparatus and a photograph of the channel during a test. The position of the current front was detected by employing the acquisition system and software described in Longo et al. In a few cases, the images were processed by hand, detecting the interface manually on the pictures when the fluid was not clearly distinguishable.
The Newtonian fluid used in the experiments was a mixture of water, glycerol and ink. Xanthan gum was added to obtain a shear-thinning fluid. The rheological behaviour of all fluids employed was independently assessed using strain-controlled rheometers coaxial cylinders or parallel plate. The temperature was measured at regular intervals during each experiment by a mercury-in-glass thermometer accuracy 0. Table 4 lists the main parameters of the tests performed in diverging channels.
Figure 7 shows experimental results compared to theoretical predictions, respectively, for linearly diverging horizontal and inclined channels with a semicircular cross section. The scaled dimensionless distance is depicted as a function of dimensionless time, expressing all factors in the various expressions of X N as f X N.
In the former case, there are several reasons that justify the discrepancy between theory and experiments. Similar problems were faced by other researchers dealing with experiments modelling an instantaneous fluid release. A preliminary indication of the time shift is given by the theoretical model, but a more correct evaluation is here provided using a best-fitting procedure with data corresponding to the intermediate-time propagation of the current. No instability was observed during the experiments.
The variables X N and T are dimensionless and scaled for each flow. The solid line represents the perfect agreement with the theory. The measurement of front position in flows of Newtonian and shear-thinning liquids have been multiplied by a factor 10 and , respectively, in order that they be separated. The problem parameters are affected by measurement or estimate errors; therefore, the propagation of uncertainty to the output variables through the variance formula allows deriving the associated confidence limits under the assumption that the input variables are independent. The individual contributions of the input parameters to the overall standard deviation of the current length are shown in figure 8 against time.
At early times, the uncertainty of the fluid index n also accounts for a significant fraction of the total uncertainty. As noted earlier, for two tests with constant flow rates 11 and 13 , the thickness of the current was measured in a single cross section; results are shown in figure 9. The left panel refers to a test in a horizontal channel, which shows a satisfactory agreement with the theory even though the two curves exhibit a time lag and a slightly different shape.
The right panel refers to the case of an inclined channel. The arrival time of the front end of the current and the maximum height are correctly predicted, the thickness of the current at late times is within the confidence interval. These tests yielded a current speed significantly slower, by a factor of 2, than that predicted by the theoretical model in this case.
We tentatively interpret these discrepancies as a demonstration that the effects of the normal stress exerted on the fluid by the walls, and acting against the flow, are significant and cannot be neglected. Further experiments are needed to support this conjecture. For horizontal converging channel, a self-similarity solution of second kind could arise [ 38 , 39 ]. This analysis is left for future work. The solid thin line is the theoretical profile with associated confidence intervals represented as dashed lines , while the bold line represents experimental points.
We have theoretically and experimentally investigated the flow of laminar gravity currents of power-law fluids in horizontal or inclined channels or fractures, with different cross-sectional shapes and longitudinally variable width. The theoretical scalings allow for the evaluation of the position of the current front and the thickness of its profile.
Laboratory experiments to validate the theory were conducted with Newtonian, shear-thinning or shear-thickening fluids in a linearly divergent channel of circular cross section. However, since a single geometry was tested, no general conclusion can be drawn on the performance of the theory, especially for the converging channels and squeezing fractures.
The dependency of these factors on model parameters is complex and governed by the influence of mass balance on one hand and of viscous resistances on the other. Some of these critical factors act as thresholds to limit the asymptotic validity of the resultant solutions to within certain ranges of parameters. Inclined channels exhibit faster currents.
Thus it is a useful model for studying mechanisms of arrhythmias post MI. In this model, 4 to 5 days after LAD occlusion there is a layer of surviving epicardium of varying thickness overlying the infarct epicardial border zone or EBZ . Ventricular tachyarrhythmias induced by programmed stimulation often result from reentrant circuits formed within the EBZ .
Although perturbations in cellular electrophysiology of the EBZ have been well characterized and linked to arrhythmogenesis, there is an important omission in the previous studies: There are two components of delayed rectifier currents: Both have been described for human atrial and ventricular myocytes [6,7] and may participate in action potential repolarization, yet relatively little is known about how disease processes affect their function and expression.
Our data showed that there was a reduction in the mRNA levels of these subunits in infarcted hearts, but the degree of reduction was not the same among the three. Experiments on animals were conducted in conformity with the Declaration of Helsinki Br Med J ; ii: The surgical procedure was as described .
Tissue from the corresponding region of normal hearts was used as a control. Ventricular myocytes were enzymatically dispersed as previously described . For NZs, only rod-shaped cells with clear striations and surface free of blebs were used for voltage clamp. IZs were chosen for study based on the following morphological criteria: About 6 min after forming the whole-cell recording WCR configuration, the bath solution was switched to a nominally Na- and Ca-free solution for 6 min, after which data collection began see Fig.
Same voltage clamp protocol as in A. I t was measured as the difference between the current level at the beginning of a voltage step and that at the end of the 5-s pulse. I tail was measured from the peak tail currents relative to the holding current. Numbers of cells and hearts studied are shown. Clamp protocol generation and data acquisition were controlled by Clampex version 5.
Membrane currents were low-pass filtered at 1 kHz Frequency Devices, Harverhill, MA and digitized at a sampling interval of 5 ms. Methods of data analysis will be described in figure legends. Clampfit of pClamp version 6. The pipette solution contained mM: Normal Tyrode's solution had mM: The nominally Na- and Ca-free solution contained mM: The pH of this solution was titrated to 7.
Total RNAs were prepared from tissues using the acid guanidinium thiocyanate phenol—chloroform extraction method .
The gel was exposed to a PhosphoImager screen and the band intensities were quantified model SI, Molecular Dynamics. Each gel contained 4 lanes of samples RNase digestion products from control hearts and 4 lanes of samples from infarcted hearts each sample from an individual animal see Figs. To correct for differences in band intensity between lanes caused by sample handling and loading, the signal from the channel subunit in each lane was normalized by that from 28S internal control in the same lane.
Since radioactive probes used in different gels were synthesized in separate reactions and had different specific activities, a second normalization procedure was required to allow data to be pooled from different gels. This was done by normalizing data values of 28S ratio with the mean value of 28S ratio from the four control hearts in each gel. Statistical analysis was performed using SigmaStat version 2. Current traces recorded from an NZ are shown in Fig.
The recording conditions were designed to eliminate interfering currents: Na and Ca currents were suppressed by the nominally Na- and Ca-free bath solution. These are hallmarks of delayed rectifier currents in cardiac myocytes [4,5]. At this concentration, azimilide blocks I Kr and I Ks with little or no effects on other K channels in canine myocytes . Under the same conditions, the same voltage clamp protocol elicited much smaller time-dependent outward currents in the IZ than those of the NZ Fig.
Furthermore, there were no clear tail currents upon repolarization. The magnitude of this time-independent current did not differ between the two. This current component was not suppressed by azimilide NZ in Fig. The identity of this current is not clear, and it is not considered in this study. Separation of I Kr and I Ks. A Currents from an NZ under the control conditions left , and in the presence of dofetilide middle or azimilide right. Tail currents recorded under the control conditions marked by the shaded area were used for the quantification of I Kr and I Ks in B. Dotted line denote zero current level.
B Control tail currents from A are shown on expanded scales.
I Kr and I Ks amplitudes were quantified from the peak tail currents vs. Data were from 19 normal hearts and 15 infarcted hearts. Each heart had one I Kr and one I Ks current density averaged from data from that particular heart 1 to 6 cells per heart. To minimize this interference, we began data collection 12—13 min after the beginning of whole-cell recording WCR, see Methods.
This result suggest that with the current setting of modeling parameters, through adjusting the ratio between the numbers of excitatory neurons to that of inhibitory neurons, the network could results in balanced synaptic current. The horizontal dashed line is corresponding to the balanced excitatory and inhibitory synaptic currents.
To demonstrate the response behavior of this network to above pulse current stimulation, we superimposed raster plots of trials onto one plot to demonstrate how this network responds to pulse inputs. In most cases, the network responds to input pulses with a burst of action potentials and then returns to a quiescent state. However, in some cases the noise would induce sustained spontaneous firings e. In this study, we assumed the information in the input signals is carried by the first wave of the spikes and used a detection window of 8 ms after the application of inputs; the window is therefore large enough to include the first wave of firings but small enough to exclude sustained and ongoing spontaneous firings Yu and Liu, In this arrangement, each neuron fires at most once within the detection window.
The thick lines at 0 ms indicate the time instant when the pulse inputs are applied. The raster plots of trials are superimposed to better show the results. As the response time of a typical HH neuron is distributed within an interval of 0—8 ms after the application of input Chen et al. In this case, larger inputs will lead to more early firings and thus a stronger inhibition current to suppress later firings.
The difference in the detection rate for strong and weak pulses again becomes trivial.
The dependence of the information capacity of the network on the ratio of the number of excitatory neurons to the number of inhibitory neurons in the network. In this range, high noise intensity will cause more neurons to fire in advance, and those firings will lead to more firings in other neurons through excitable connections. Therefore, a higher noise intensity increases firing probability and energy cost. Energy efficiency is defined as the ratio of mutual information to the energy cost in response to input pulses.
This implies that there exists an optimal ratio of excitatory to inhibitory connections in the network at which the amount of information transmitted is maximized for a given unit energy cost. Because high noise intensity will sabotage information processing in the network, as demonstrated in Figure 4B , energy efficiency decreases as the noise intensity increases.
The vital role of fixed energy in the maximization of energy efficiency has been reported in several previous studies Schreiber et al. They found that the energy efficiency decreases monotonically as the system size increases if the fixed energy cost is not taken into consideration. Therefore, to maximize energy efficiency, the fixed energy cost should be approximately the same order of magnitude as the energy cost of information processing. In the previous section, we have obtained the analytical solution for the network dynamics Equation 9 , information transmission Equation 11 , as well as the energy cost Equation 12 of the bistable neuron network as a function of net synaptic current.
Figure 6A demonstrated the dependence of pulse signal detection rate on the net synaptic current for different input signal strengths. The analytic solution for a bistable neuron network. A The dependence of the pulse signal detection rate on the net synaptic current for different input signal strengths. C The total energy cost of a neuronal network as a function of the net synaptic current for different noise intensities. E The energy efficiency as a function of the net synaptic current for different fixed energy costs. F The energy efficiency as a function of the net synaptic current for different energy costs per synaptic event.
As shown in Figure 6C , when the net synaptic current is larger than zero, the energy cost increases as the net synaptic current increases, and higher noise intensities result in higher energy costs. This result is in accordance with the simulation results. Because of the over-simplified consideration of inter-neuronal connections in the simulation, our analytical result fails to replicate these behaviors. Figures 6E,F show that the energy efficiency decreases as the fixed energy cost or the energy cost for synaptic activity increases. The optimal net synaptic current for maximal energy efficiency moves to the left with an increased fixed energy cost and moves to the right with an increased synaptic energy cost.
Thus, our analytical solution is consistent with the simulation results described above. In this study, we investigated the dependence of the information transmission and energy efficiency of a neuronal network on the balance of excitatory and inhibitory synaptic currents through both computational simulation of classical HH neurons and analytic solution of bistable neurons with a mean-field approximation.
Many studies have found advantages to neuronal networks for organizing around this critical point, such as maximized information storage Haldeman and Beggs, , maximized dynamic range Shew et al. As argued in Shew et al. Our results also demonstrate that through stochastic resonance phenomenon in which the optimal noise intensity maximizes the information transmission of a nonlinear threshold system, the noise could enhance the information transmission in the network Figures 4B—D , 6B. In previous decades, many studies have shown that to process information in an energy efficient manner, neuronal systems optimize their morphological and physiological parameters, e.
They found that spikes evoked by balanced synaptic currents are more informative and energy efficient Sengupta et al. Recent work also revealed that the cost-efficient information capacity with minimal spike rate can be achieved in the regime of moderate synchrony, where the irregular firing, synchronized oscillations and neuronal avalanches can be observed simultaneously Yang et al.
Therefore, our results, along with those of other studies, demonstrate the possibility that neural systems may optimize their morphological and physiological parameters to be energy efficient. However, though balanced excitation-inhibition network often leads to critical-state dynamics Poil et al. Our current work did not take into account of the nontrivial dynamics patterns and its interaction with firing rates, which is left for our future study.
The dependence of the maximization of energy efficiency on the fixed energy cost has been reported in several studies. For example, both in a single neuron with graded potentials and a neuronal population, the maximization of energy efficiency by the number of ion channels or the number of neurons requires the inclusion of a fixed energy cost Schreiber et al.
Here, we argue that this fixed energy cost could be assigned to the cost of generating spontaneous firings and consequent synaptic activity due to noise perturbation. In our calculation, the costs directly related to signal processing the energy consumed by action potentials invoked by input signals and synaptic transmission are explicitly calculated.
However, spontaneous firing is an ongoing process that continually costs energy even without input signals. Normally, this energy cost is a constant within a unit time interval, assuming the spontaneous firing rate is a constant. Therefore, a larger fixed energy cost can be considered a longer interval within which no inputs are applied. Therefore, we speculate that to maximize energy efficiency, the signal input rate must be below a certain threshold so that a sufficient fraction of the fixed energy cost can be accounted for in the total energy cost.
The energy cost of synaptic events greatly affects the energy efficiency of a network. In this study, we expanded our previous solution of the response function for a bistable neuron from a single isolated neuron to neurons with excitatory and inhibitory synaptic connections by adding a modification term to represent the effects of a net synaptic current and noise on the firing probability of a neuron to pulse inputs. This mean field approximation is a simplification of the complex interactions between neurons, although in some cases, it fails to replicate the exact behavior of our simulation results e.
Therefore, we expect that a more explicit form of this interaction term would lead to a more accurate description of the average response of the neurons in a network and would improve our understanding of the dynamics of neuronal networks with excitatory and inhibitory connections. This will be a direction for our future research. It is noticed that previous work suggested that through tuning the ratio of excitatory to inhibitory synaptic current intensity, the network could be well balanced to maximize the energy efficiency Shew et al. This result suggests that a certain ratio e. We have to point out the limitation of our work.
First, information is not only carried by the firing rate, but also in the spike-timing interval as well as the population correlations Panzeri et al. Second, in real cortex, there are multiplex network configurations, and most of them are sparsely connected. However, our analysis and results are derived from a fully recurrent connected network configuration, which might only provide a linear-style understanding on the principles of the cortical network organization.
In this work, we used a uniform recurrent network structure that neurons are all-to-all connected. Recent work demonstrated that structural heterogeneity in cortical network could undermine the balanced state, while homeostatic synaptic plasticity can recover the balance of network excitation-inhibition Landau et al.
Therefore, more studies are needed to test if the conclusion obtained here still hold in the biologically more realistic case. These results are further confirmed by an analytical solution for a bistable neuron in which interconnection between neurons is approximated with a mean-field approach. These results reflect general mechanisms for sensory coding processes that may provide insight into energy efficient neural communication and coding.
All authors reviewed the manuscript. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Energy-efficient action potentials in hippocampal mossy fibers. An energy budget for signalling in the greymatter of the brain. Rosenblith Cambridge; Massachusetts, MA:
In the former case, there are several reasons that justify the discrepancy between theory and experiments. This decrease almost certainly reflected the clogging of the pipette tip by the O 2. For squeezing fractures, the analysis is more complex as a second threshold b t becomes important, depending on the channel shape k. In very fast-changing fields, the magnetic field does not penetrate completely into the interior of the material. Another example involves dropping a strong magnet down a tube of copper  — the magnet falls at a dramatically slow pace. Recent work also revealed that the cost-efficient information capacity with minimal spike rate can be achieved in the regime of moderate synchrony, where the irregular firing, synchronized oscillations and neuronal avalanches can be observed simultaneously Yang et al.