Charlotte Héricé

Radwa Khalil

Maria Moftah

Thomas Boraud

Martin Guthrie

André Garenne

Decision-making in a neural network model of the basal ganglia

BEHAVIORAL TASK

We submit the model to a protocol 2,3 for BG involvement in decision-making with monkeys in conditions of uncertainty. There are 4 different cue shapes, each with its own reward probability and 4 possible positions. At each trial : 1. Random presentation of 2 cue shapes (at random positions) 2. Choice made by the monkey and the model 3. Reward given or not according to the reward probability of the shape

CONCLUSION

We have presented here, for the first time, a biophysically based, spiking neuron model of the BG that is able to perform 2 levels action selection. This model is closely based on the known anatomy and physiology of the basal ganglia and demonstrates a reasonable mechanism of network level action selection.

This cellular and synaptic level of description bridges the gap between top-down mesoscopic level of description1 and a bottom-up approach relying on emerging properties of neuronal networks dynamics. Our model is also able to predicts some important behavioral characteristics like localized lesion consequences on learning impairment and intrinsic dynamics, reversal learning and extinction protocol.

Decision-making in a neural network model of the basal ganglia

→

  Probabilistic learning task →T h e m o n k e y a n d t h e model both have to learn to chose the optimum cue shape (the one with the best reward probability). INTRODUCTION Basal ganglia (BG) are known to host mechanisms of action selection and its adaptation to a changing environment. Their architecture consist of several parallel functional loops connecting back to distinct areas of cortex (motor, cognitive and limbic) and processing different modalities of decision making. The picture of parallel loops is complicated by partial convergence and divergence connections that implies that the various loops are interacting. A previous BG model 1 was built of interacting bloc-diagram based on rate-models. It was able to learn optimized action selection during a probabilistic reward task. The aim of the present work is to refine and extend these results to a cellsynapse level through a bottomup approach. → Highlighting of the structure-function relationship and circuitry emerging properties. → Investigation of cell-scale mechanisms impact on the whole model capacities (learning and decision-making).

M. Guthrie et al. Interaction between cognitive and motor cortico-basal ganglia loops during decision making: a computational study,

J. NeuroPhysiology

Charlotte HÉRICÉ 1, 2, 3 , Radwa KHALIL 1 , Marie MOFTAH 4 , Thomas BORAUD 1, 2 , Martin GUTHRIE 1, 2 and André GARENNE 1, 2, 3 Contact: charlotte.herice@u-bordeaux.fr andre.garenne@u-bordeaux.fr

→ With the time course of the average firing rate (A), we are able to see the evolution of motor and cognitive cortex for example. → A decision is made when a difference in the activities more than 40 Hz is observed. → The higher activity represents the choice

Good Choices Good Decisions

Time (ms)

Average Firing Rate (Hz)

Average Firing Rate (Hz)

Cognitive decision Motor decision

A.

RESULTS

-Exploration: An expected emergent property of the network is a divergence in the cortical activations of cognitive and motor sub-populations. In the absence of learning the network is still able to make a decision. This is equivalent to decision-making during the exploration phase of reinforcement learning.

-Exploitation: A Good Choice (GC) is made when the optimal shape is selected and a Good Decision (GD) when the associated direction is selected too. Both are improved during a standard learning session (B).

During training, the model learns to create a dynamic link between the cognitive and motor sensory component of a cue. This can be assessed by the learning curves profile of the model (C).

→ The average reward and GC rate gradually increase along the session (C). → The optimum cue shape direction is preferentially selected (B). → The movement onset delay is decreased by the learning (D).


Clifford W Fong
email: cwfong@internode.on.net

Anti-cancer properties of "hydrophilic and hydrophobic" statins and cyclin-dependent kinases

Keywords: Statins, anti-cancer properties, QSAR, cyclin-dependent kinases

The general QSAR equation: Transport or Inhibition = ΔG desolvation + ΔG lipophilicity + Dipole Moment + Molecular Volume is shown to apply to statin inhibition of HMG-CoA reductase, passive and active statin cellular transport, the inhibition of growth of pancreatic cancer cells, and the survival times of mice. The equation also applies to the inhibition of cyclindependent kinases, which are known to be involved with statins in affecting the progression of tumour cell cycles.

The common clinical practice of using "hydrophilic" and "hydrophobic" statins as being arbiters of anti-cancer efficacy is clearly wrong. Efficacy of statins is a function of how well desolvation, lipophilicity, dipole moment and molecular volume of individual statins affect the many pathways involved in their anti-cancer activities.

Abbreviations

ΔG desolvation free energy of water desolvation, ΔG lipophilicity free energy of lipophilicity or hydrophobicity, DM dipole moment, R 2 multiple correlation coefficient, F the F test of significance, SEE standards errors for the estimates, SE(ΔG desolvation ) standard errors of ΔG desolvation , SE(ΔG desolv,CDS ) standard errors of ΔG desolv,CDS , SE(ΔG lipophilicity ), standard errors of ΔG lipophilicity , SE(Dipole Moment) standard errors for dipole moments, SE (Molecular Volume) standard errors for molecular volumes as calculated from "t" distribution statistics

Introduction (a) Statins anti-cancer properties

Statins or 3-hydroxy-3-methylglutaryl coenzyme A (HMG-CoA) reductase inhibitors are currently the most commonly prescribed drug class in the world. Statins are used for primary and secondary prevention of cardiovascular diseases. Besides cholesterol reduction, preclinical studies have shown that statins may exert anti-neoplastic effects, through both HMG-CoA reductase-dependent and HMG-CoA reductase-independent pathways. By competitive inhibition of HMG-CoA reductase in the mevalonate cycle, (Figure 1) statins prevent a variety of pathways that lead to the inhibition of tumour cell growth, metastases, angiogenesis and apoptosis. Statins are known to have an effect on the cell cycle particularly on the cyclin-dependent kinases (CDK) that regulate the cell cycle. [START_REF] Matusewicz | The effect of statins on cancer cells -A review[END_REF][START_REF] Gazzerro | Pharmacological Actions of Statins: A Critical Appraisal in the management of Cancer[END_REF][START_REF] Singh | Statins -the Holy Grail for cancer?[END_REF][START_REF] Altwairgi | Statins are potential anticancerous agents[END_REF][START_REF] Hindler | The Role of Statins in Cancer Therapy[END_REF][START_REF] Moon | Statins: protectors or pretenders in prostate cancer?[END_REF] (Figure 2) It is known that lovastatin at a concentration of 5 μM induces breast cancer cells to arrest in the G1 phase by inhibiting the proteasome which, in turn, leads to the accumulation of inhibitors of cyclin-dependent kinases (CDKI), such as p21 WAFI/CIP1 and p27. Lovastatin lactone is responsible for the inhibition of the proteasome and thus inhibition of p21 WAFI/CIP1 and p27 degradation. It has been proven that mevalonate restored and increased the activity of proteasome, which caused degradation of CDKI and facilitated entry into the next phase of the cell cycle. [START_REF] Rao | Lovastatinmediated G1 arrest is through inhibition of the proteasome, independent of hydroxymethyl glutaryl-CoA reductase[END_REF] There are very few agents available to prevent the onset of cancers. Statins have been shown to be safe in humans and widely used clinically for lowering cholesterol levels. So, if they prove to have potent anti-cancer effects, it will be easy to adapt them for prevention and therapy of cancers in humans. HMG-CoA reductase inhibition by statins leads to anti-cellcycle inhibition of cyclin-dependent kinases and G1 to S progression, as well as additional inhibition of other signal transducers.

However while there is evidence supporting the positive role for statins in preventing cancer, there are some contrary views that statins may in fact cause cancer. 8 A major difficulty in clarifying the role of statins regarding their anti-cancer behaviour in clinical trials is that there are very significant confounding issues that cloud the interpretation of clinical outcomes, such as age, many patients having existing cardiovascular disease, diabetes mellitus, being users of aspirin, metformin and other concomitant drugs, smokers, statin users being health conscious and more compliant with cancer screening leading to early cancer detection and treatment, translating improved survival etc. 3 Two recent studies indicate that statin use was associated with an overall 22% reduction in deaths from breast, ovarian, colorectal, digestive cancer types (but not lung cancer) in a large number of post-menopausal women over 15 years and a 55% reduction in deaths from bone/connective tissue cancers. [START_REF] Aragaki | Statin use and all-cancer mortality: Prospective results from the Women's Health Initiative[END_REF] A study in a large number of men looked at statin use together with the anti-diabetes medication metformin and found a 40% reduction in prostate cancer mortality, with the effect more pronounced in men with obesity/metabolic syndrome. [START_REF] Lu-Yao | A population based study[END_REF] There are several mechanisms in cancer cells that are activated by statins, some of them leading to cell death. [START_REF] Matusewicz | The effect of statins on cancer cells -A review[END_REF][START_REF] Gazzerro | Pharmacological Actions of Statins: A Critical Appraisal in the management of Cancer[END_REF][START_REF] Singh | Statins -the Holy Grail for cancer?[END_REF][START_REF] Altwairgi | Statins are potential anticancerous agents[END_REF][START_REF] Hindler | The Role of Statins in Cancer Therapy[END_REF][START_REF] Manu | Simvastatin sensitizes human gastric cancer xenograft in nude mice to capecitabine by suppressing nuclear factor-kappa B-regulated gene products[END_REF][START_REF] Jiang | Novel anti-glioblastoma agents and therapeutic combinations identified from a collection of FDA approved drugs[END_REF][START_REF] Han | A randomized phase II study of gefitinib plus simvastatin versus gefitinib alone in previously treated patients with advanced non-small cell lung cancer[END_REF][START_REF] Held | Genotypeselective combination therapies for melanoma identified by high-throughput drug screening[END_REF] Statins exert different anti-cancer effects depending on:

• cell line and tumour type: clinically statins are considered to be effective against ovarian, colorectal, digestive cancer types (but not lung cancer) in post-menopausal women over 15 years and a 55% reduction in deaths from bone/connective tissue cancers. [START_REF] Aragaki | Statin use and all-cancer mortality: Prospective results from the Women's Health Initiative[END_REF] In men, statin use together with the anti-diabetes medication metformin, a 40% reduction in prostate cancer mortality was found. The US National Cancer Institute is investigating the role of statins in colorectal and melanoma skin cancers. • statin species and concentration: statins can exist in three possible forms, the neutral acid, the neutral lactone, and the anionic acid form, with concentrations depending on pH, and whether some statins are administered as the lactone pro-drug eg lovastatin and simvastatin. However small concentrations of the three forms can exist in equilibrium around the physiological pH. • the type of statin ("hydrophilic" or "hydrophobic") may have a large effect on the transport of statins across the cell membrane. The form of the statins has a large effect on the hydrophobic and hydrophilic properties of statins, with the neutral acid and lactones being more hydrophobic than the charged anionic form which are more hydrophilic. • duration of exposure of tumour cells to statins, such as statin influx into cells and statin efflux out of the cell, particularly by multi-drug resistant transporters, and metabolic clearance of statins • by inhibition of the synthesis of cholesterol, statins may destabilize the cell membrane by reducing myocyte membrane cholesterol. Instability of the cell membrane could lead to cell lysis and myopathy. Reduction of cholesterol could affect transport into and out of the cell, possibly by changing the fluidity of the lipid rafts in the cell membrane. • statins may change the arrangement of the transporter OATP1. OATP1B1 is the major transporter in hepatocytes.

• statins may affect the localization of HMG-CoA reductase • statins can inhibit the cell cycle by affecting both expression and activity of proteins involved in cell-cycle progression such as cyclins, cyclin-dependent kinases (CDK), and/or inhibitors of CDK. • statins may induce apoptosis by both intrinsic and extrinsic pathways • statin treatment may lead to changes in molecular pathways dependent on the EGF receptor, mainly via inhibition of isoprenoid synthesis • statins may induce conformational changes in glucose transporter proteins • statins will be used in combination with other anti-cancer drugs in chemotherapeutic regimes

In summary, it is clear that the anti-cancer efficacy of statins is dependent on many cellular pathways and the physico-chemical and metabolic properties of the various statins.

(b) Hydrophobic versus hydrophilic statins: anti-cancer efficacy

Much of the focus on the preventative in vivo anti-tumour effects of statins has centred on whether there is a positive anti-neoplastic effect that can be distinguished from the many confounding comorbiditity effects. Little attention has focussed on what properties of statins may be important for anti-tumour effectiveness. The principal differentiator in the literature focuses on the "lipophilic" or "hydrophobic" statins as being effective, and the "hydrophilic" statins as being less effective. [START_REF] Matusewicz | The effect of statins on cancer cells -A review[END_REF][START_REF] Sivaprasad | Differential efficacy of 3-hydroxy-3-methylglutaryl CoA reductase inhibitors on the cell cycle of prostate cancer cells[END_REF][START_REF] Feldt | Statin-induced anti-proliferative effects via cyclin D1 and p27 in a window-of-opportunity breast cancer trial[END_REF][START_REF] Brown | The differential effects of statins on the metastatic behaviour of prostate cancer[END_REF][START_REF] Kato | Lipophilic but not hydrophilic statins selectively induce cell death in gynaecological cancers expressing high levels of HMG CoA reductase[END_REF][START_REF] Duncan | Statins and cancer[END_REF][START_REF] Coogan | Statin use and the risk of 10 cancers[END_REF] Hydrophilic statins are considered to be pravastatin and rosuvastatin, while lipophilic or hydrophobic statins are considered to be cerivastatin, simvastatin, lovastatin, fluvastatin, atorvastatin, and pitavastatin. Hydrophilic statins are thought to mainly accumulate in the liver through uptake by the hepatic transporter OATP1B1, whereas hydrophobic statins are thought to be widely distributed in various tissues as they are considered to be able to passive permeate through cell membranes, unlike the hydrophilic statins. Hydrophilic statins are mainly metabolized by cytochromes, whereas pravastatin and rosuvastatin are not metabolically cleared by cytochromes.

However the notion that hydrophilic statins cannot easily pass through cell membranes (unless they are actively transported by hepatic transporters such as OATP1B1) unlike the hydrophobic statins is not supported in studies of whether statins can pass through the blood brain barrier (BBB), the most physiologically tight barrier in the body. There is evidence that hydrophilic and hydrophobic statins can pass through the BBB. Simvastatin, lovastatin and pravastatin (supposedly hydrophilic) have been detected in the cerebral cortex, with brain levels of the three statins reflecting their lipophilicity: simvastatin levels >lovastatin levels> pravastatin levels. Pravastatin has been shown to be transported into cells by organic anionic transporters (OATs) and these transporters are expressed in the brain. Another transporter that transports statin acids into cells is the monocarboxylic acid transporter, MCT4. In a study of simvastatin and lovastatin lactones and acids, it was concluded that the lactones were transported across the rat BBB by diffusion, whereas the acid forms were thought to be transported by MCT transporters. [START_REF] Shepardson | Cholesterol Level and Statin Use in Alzheimer Disease II. Review of Human Trials and Recommendations[END_REF][START_REF] Wood | Statins and Neoruprotection: basic pharmacology needed[END_REF][START_REF] Kobayashi | Role of Monocarboxylate Transporter in Statin-induced Cytotoxicity[END_REF][START_REF] Tsuji | Transport mechanism of 3-hydroxy-3methylglutaryl conenzyme A reductase inhibitors at the blood-brain barrier[END_REF] Statins can exist in three forms, often in equilibrium depending on the physiological conditions, particularly pH: the neutral acid form, the charged anionic acid form, and the neutral lactone form. Simvastatin and lovastatin are administered as the lactone, which then undergo biotransformation to the acid and then the anionic form around the physiological pH 7 to act as inhibitors of HMG-CoA reductase. The role of the lactone form during various metabolic processes, including cytochrome clearance, has often been under recognized, [START_REF] Fong | Statins in therapy: Cellular transport, side effects, drug-drug interactions and cytotoxicity -the unrecognized role of lactones[END_REF] but statin lactones play a critical role in muscle myopathy and related unwanted side effects of statins. The form of the statins plays a crucial role in transport of statins across cell membranes, including the fact that the simvastatin and lovastatin lactones are the actual species transported across the BBB. [START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3-hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] The notion that the anti-cancer ability of statins can be differentiated into cell membrane permeable hydrophobic and non-permeable hydrophilic statins is clearly oversimplified and largely incorrect. The notion that statins can be either hydrophilic or hydrophobic is also misleading and incorrect, since at a molecular level statins have both hydrophilic and hydrophobic (lipophilic) areas on the molecular surface exposed to water (or proteins when bound to that protein). The much used measure of lipophilicity logP for unionized drugs (or logD for ionized drugs) is based on partitioning of a drug between water and n-octanol, with n-octanol representing the lipophilic phase. However, n-octanol contains 2.8M water content, which can have a large influence on dissolution into the lipophilic phase, especially for drugs that are not very water soluble (most drugs). Hence logP or theoretical methods based on logP can be a misleading criterion for drug lipophilicity or hydrophobicity. [START_REF] Fong | Statins in therapy: Cellular transport, side effects, drug-drug interactions and cytotoxicity -the unrecognized role of lactones[END_REF][START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: causes of resistance[END_REF][START_REF] Fong | Drug discovery model using molecular orbital computations: tyrosine kinase inhibitors[END_REF][START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3-hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] The form of the statin species has a large effect on the hydrophilic or hydrophobic properties of the statins, and all three acid, acid anion and lactone forms can be present in varying quantities in equilibrium around the physiological pH level.

(c) Application of QSAR model to predict statin anti-cancer behaviour

This study will use a model (general QSAR equation) which has been successfully applied to various transport and drug-protein binding processes. A common model for both processes based on 4 molecular physico-chemical properties of the prospective drugs has been previously applied to passive and facilitated diffusion, and active organic anion transporter drug membrane transport, statins-HMG-CoA reductase inhibition, some competitive statin-CYP enzyme binding processes, tyrosine kinase inhibitors, HIV-1 protease inhibitors, and cyclin-dependent kinases. The general QSAR equation also applies to the active competitive transport of these tyrosine kinase inhibitors by the hOCT3, OATP1A2 and OCT1 transporters. [START_REF] Fong | Statins in therapy: Cellular transport, side effects, drug-drug interactions and cytotoxicity -the unrecognized role of lactones[END_REF][START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: causes of resistance[END_REF][START_REF] Fong | Drug discovery model using molecular orbital computations: tyrosine kinase inhibitors[END_REF][START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3-hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] The model comprises four main properties: (a) desolvation energy in water (b) lipophilicity or hydrophobicity based on the solvation energy in a hydrophobic solvent such as n-octane or n-octanol; lipophilicity being a measure of how well a drug can interact with lipophilic cell membrane bilayers, and hydrophobicity being a measure of non-polar interaction between a drug and the hydrophobic sectors of a protein (c) dipole moment in water, as a measure of the polar attraction between the drug and its receptor target or cell membrane (d) the molecular volume of the drug in water as a measure of how well the prospective drug fits into the cavity of the target receptor protein or active protein transporter, or how well a drug can passively diffuse through a cell membrane. The basic model is shown in the general QSAR equation.

= desolvation + lipophilicity + $ %& ' (& + ' %&)*% + %*(&

Results

(a) Inhibitory binding of statins to HMG-CoA reductase

Corsini et al [START_REF] Corsini | Pharmacology of competitive inhibitors of HMG-CoA reductase[END_REF][START_REF] Bellostra | History and development of HMG-CoA reductase inhibitors[END_REF] have studied the inhibition of HMG-CoA reductase activity K i (nM) by statin acid anions. These data show reasonable correlations with the general QSAR equation.

Inhibition of rat microsomal HMG-CoA reductase activity K i (nM) by statin acid anions 30 [Corsini 1995 Equations 1 is highly significant, and indicates the most sensitive binding variable is the dipole moment, which are high for the anionic species. The desolvation energy, lipophilicity and molecular volume of the statins are of about equal sensitivity. Molar Volumes have been multiplied by 0.1 to give magnitude range similar to the magnitude ranges of other three independent variables in applying the general QSAR equation. Rosuvastatin is a very large outlier from the correlations, with mevastatin also a significant outlier. Rosuvastatin activity is sourced from a different study [31] than the other statin activity data from the one study. [30] Inclusion of mevastatin in eq 1 still gives a good correlation, R 2 0.877, F significance 0.100 but it is clear that eq 1 is a stronger correlation if mevastatin is treated as an outlier. Correlations of activity were tested against the statin acids, but the correlations were poor, indicating that the anionic species which are the dominant species at physiological pH, are the likely inhibitory species. Equation 2a which shows the passive diffusion rates for statin lactones in rat hepatocytes can be compared directly with eq 2 showing the in vitro passive diffusion rates for the same lactones. It can be seen that the relative magnitude and signs of the coefficients are similar (though the in vitro eq 2 is ca 10 times more sensitive than the in vivo eq 2a), with the exception of the dipole moment which is relatively much larger in the rat hepatocyte than the in vitro case.

An analysis of the published K m rat data from the same experimental study is available for rosuvastatin 7.5, pitastatin 6.3, pravastatin 30.0, fluvastatin 37.6, atorvastatin 4.0 and cerivastatin 7.0 μM. [START_REF] Shitara | Clinical significance of organic anion transporting polypeptides (OATPs) in drug disposition: their roles in hepatic clearance and intestinal absorption[END_REF] This data has been analysed in equations 4 and 5. Equation 4 (lactone form) is more statistically robust than equation 5 (acid anion form) which suggests that it is the lactone form that is preferentially taken up by the active OATP transporter, and it is the lipophilicity and dipole moment that governs the thermodynamic binding affinity between the statin lactone and the OATP transporter. This observation is consistent with an essentially electrostatic interaction plus a hydrophobic interaction between the polar and non-polar portions of the statin and the transporter protein respectively. The K m correlation with the lipophilicity and dipole moment is consistent with a substrate-protein equilibrium where the substantially desolvated statin substrate lies within, and interacts with the protein environment, and outside the bulk blood plasma environment. If equation 4 and 5 are valid then a substantial desolvation must occur before the statin can enter the protein environment and bind to it, and that the hydrophobic interaction is counterbalanced by the electrostatic interaction in the total binding interaction.

However these indicative equations (which are not strongly robust in view of the limited number of experimental data points) can only be supportive evidence, but data from other literature studies does support the hypothesis that the lactone form is the likely form that is responsible for the majority of the passive diffusion and active transport by OATP transporters. It is possible that both neutral acid and lactone forms are competitively transported across the cell membrane. In view of the molecular volume or size of these statins, it is likely that the transport mechanism is a facilitated passive diffusion process.

For the statin lactones: Corning Transporto cells, [START_REF] Li | Use of a Novel Cell-based Model to Predict Hepatic Clearance of Statins: In Vitro to In Vivo Correlation, Corning[END_REF] Eq 2 and eq 2a are similar, indicative of similar (probably facilitated) diffusion processes.

For the statin acids: Corning Transporto cells, [START_REF] Li | Use of a Novel Cell-based Model to Predict Hepatic Clearance of Statins: In Vitro to In Vivo Correlation, Corning[END_REF] For the active transport of statin acid anions by OATP transporter, rat hepatocytes, [START_REF] Shitara | Clinical significance of organic anion transporting polypeptides (OATPs) in drug disposition: their roles in hepatic clearance and intestinal absorption[END_REF] The correlation for the active transport of statin acids by OATP transporter, rat hepatocytes, [34] was very poor, suggesting that OATP transporters do not transport the acid species. Eq 4 and 5 are indicative of rat OATP transporters being active with both anion and lactone species.

(c) Anti-cancer activity of statins: in vitro and in vivo

An elegant and extensive study of the inhibitory effects of statins on the growth of in vitro pancreatic cancer cells and the in vivo effect on nude mice survival has been documented. [34] These data have been analysed by applying the general QSAR equation, and the results are given in Table 1. The values for the properties are the coefficients for the correlation equations derived from the general QSAR equation and the values in brackets are the standard errors for the coefficients. R 2 is the square of the multiple regression coefficient, F significance is the F test of significance for the multiple regression correlation equation. Molar Volumes multiplied by 0.1 to give magnitude range similar to the magnitude ranges of other three independent variables for comparative purposes

Table 1 shows the analyses using equation 1 for the inhibition of growth of three pancreatic cancer cell lines as well as the mice survival times for seven statins, rosuvastatin, pravastatin, fluvastatin, atorvastatin, cerivastatin, lovastatin and simvastatin (as acid anions). [START_REF] Gbelcova | Differences in antitumor effects of various statins on human pancreatic cancer[END_REF] The statistical analyses are not strongly robust as there are insufficient experimental data points, but the patterns across the three cell lines and mice data are consistent across cell lines and time. The standard errors of the coefficients indicate the statistical precision. The relative changes across cell lines and with time as well as the live mice results are critical in trying to elucidate mechanistic factors from this study. Gbelkova 34 has indicated that other possible statin species (such as lactones) are not involved, and the known pK a of the statins under the experimental conditions indicate that the anionic form of the free acids are the active statin species.

Examination of the results in Table 1 show consistent patterns for the three in vitro pancreatic cancer cell lines: 1. Desolvation, dipole moment and molecular volume of the statins are of similar importance in inhibiting the growth of the cells. Lipophilicity has a minor effect. 2. There is a clear time elapsed effect, particularly for the effect of dipole moment. The time effect is most clear for the BxPc-3 cells where the desolvation, lipophilicity and dipole moment influence become consistently more positive over time. This suggests that the overall reaction mechanism of inhibition is changing. 3. The nude mice survival times are reflective of the cell line results (in that the relative magnitudes of the coefficients for desolvation, lipophilicity and dipole moment to each other are similar to those from the cell line data) with the exception of the much greater negative effect of molecular volume of the statins on mice survival times. This outcome suggests that transport of the statins to the mouse tumours, either by passive or active transport mechanisms, is a critical factor. Molecular volume has a strong influence on the passage of molecules through the cell membrane bilayer in passive permeation, as well as a major effect on active transport across cell membranes via transporters, as shown in eqs 2-4.

(d) Properties of cyclin-dependent kinases (CDK)

We have previously studied the inhibitory activity against CDK2/cycE by the series Similar correlations to eq 6 were found for CDK1/cycB and CDK4/cycD.

Discussion

It is clear from the literature that the anti-cancer properties of statins are multi-causal, and that there are many inter-linked pathways that contribute to the overall efficacy of statins. It is implausible that all these pathways can be identified or separate mechanisms portrayed.

However, if a "universal" QSAR can be identified that may apply to all or most of these pathways, then some degree of predictability may be possible. It is postulated that the general QSAR equation may be applicable, since it appears to apply to many binding and cellular transport activities for a wide range of drugs.

Critical factors that govern the statin efficacy include bioavailability, the potential equilibria that exists amongst the acid, acid anion and lactone forms of the statins in the circulatory system at the physiological pH and within the tumour cell at the cystolic pH (usually slightly acidic), the rate that the various forms of the statins can be transported across the cell membrane in tumour tissue (influx), the rate of efflux out of tumour cells, metabolic clearance rates, followed by the rate of the HMG-CoA reductase inhibition by the statins, and the rates of the various other anti-cancer inducing pathways described in Figure 1. A logical and first order approach to the plethora of processes involved is to treat the influx rate into the tumour cell as the overall rate determining step that governs all other sequential processes that occur within the cell, including efflux. The second fastest rate determining step should be the rate that statins inhibit HMG-CoA reductase in tumour cells. If these two assumptions are predominantly correct, then the form of the general QSAR equation for these two steps should be a rough approximation of the overall process. Eq 1-5 are broadly consistent with this notion. Analysis of the Table 1 data suggests that the 24-72 hour cell data for the three cell lines changes and appears to start to mirror the mice survival results in that the relative magnitudes of the desolvation, lipophilicity and dipole moment coefficients are similar, with the exception of the much stronger dependence on molecular volume for the mice data. This result may indicate that transport of the statins into the tumour cells is the long term dominant determinant of anti-cancer efficacy.

The general QSAR equation has been shown to apply to inhibition of HMG-CoA reductase and the passive and active transport of statins, as shown in eq 1-5. It also applies to the inhibition of a wide range of cyclin-dependent kinases (eq 6), which is significant since it is known that there exists some relationship amongst statins, CDK and cell cycle progression.

The analysis of the effect of statin properties on growth of pancreatic cancer cells and nude mice survival times shown in Table 1 clearly show that the general QSAR equation can accurately describe how statins affect pancreatic cancer cell growth and mice survival times. The cell line data also shows a time dependency, which would be expected for multiple pathways that change in importance as a function of time.

Finally, the application of the general QSAR equation to statin anti-cancer efficacy clearly shows that the commonly used clinical distinction between "hydrophilic" and "hydrophobic" statins as being arbiters of anti-cancer efficacy is clearly wrong. Efficacy of statins is a function of how well desolvation, lipophilicity, dipole moment and molecular volume of individual statins affect the many pathways involved in their anti-cancer activities.

Conclusions

The general QSAR equation: Transport or Inhibition = ΔG desolvation + ΔG lipophilicity + Dipole Moment + Molecular Volume is shown to apply to statin inhibition of HMG-CoA reductase, passive and active statin cellular transport, the inhibition of growth of pancreatic cancer cells, and the survival times of mice. The equation also applies to the inhibition of cyclindependent kinases, which are known to be involved with statins in affecting the progression of tumour cell cycles.

The common clinical practice of using "hydrophilic" and "hydrophobic" statins as being arbiters of anti-cancer efficacy is clearly wrong. Efficacy of statins is a function of how well desolvation, lipophilicity, dipole moment and molecular volume of individual statins affect the many pathways involved in their anti-cancer activities.

In the absence of better predictive methods, the general QSAR equation may serve as a useful predictor of overall statin anti-cancer efficacy and some of the many pathways that contribute to the overall efficacy. Footnotes to Table 2:

Solvation energies are calculated using the SMD -Polarizable Continuum Model (IEFPCM), Unified Force Field, scaled van der Waals surface cavity. Solvation (free) energies are the differences between the energies of the optimised statin in the gas phase and in the particular solvent. A. V. Marenich, C. J. Cramer, and D. G. Truhlar, J. Phys. Chem. B, 2009, 113, 6378-96. Lipophilicity is the solvation free energies in n-octane.

Values in cm [START_REF] Singh | Statins -the Holy Grail for cancer?[END_REF] /mol are molecular volumes in water defined as the volume inside a contour of 0.001 electrons/Bohr 3 density. The angles 90 o etc refer to the conformational angles between the 4-FC6H5-group and the relevant heterocyclic ring where applicable.

Figure 1 .

 1 Figure 1. Schematic diagram of inhibition of mevalonate pathway by statins, and anti-tumour effects of statins

Figure 2 .

 2 Figure 2. Schematic diagram showing relationships amongst the various cell phases, cyclins, cyclin-dependent kinases (CDK) and how statins disrupt expression/activation of cell cycle genes/proteins. Asterisks (*) indicate not in prostate cancer cells, arrows pointing down (↓) indicate decrease, and arrows pointing up (↑) indicate increase. X inhibition. (Adapted from Matusewicz 2015)

( b )

 b Application of the general QSAR model to passive and active transport of statins across cell membranes (b)(1) Passive transport of statins A passive diffusion permeation model for atorvastatin, fluvastatin, cerivastatin, pravastatin, pitastatin, lovastatin, rosuvastatin statins based on passive permeation diffusion data for the Corning Transporto cells control cells 32 has been developed. The best fit of the passive diffusion rate was for the form, where equation 2 shows a positive correlation with the desolvation free energy in water, dipole moment and molecular volume, and a negative correlation with the lipophilicity.

  b]pyridin-5-yl)(phenyl)methanone (Figure3) where RO was the alkoxy group with R being n-propyl (IC 50 =0.18 μM), n-butyl (0.11), 1-methylbutyl (0.24), 3methylbutyl (0.18), n-pentyl (0.52), 2-hydroxyethyl (3.7), 2-dimethylaminoethyl(17), cyclohexyl (0.66), benzyl (0.28), phenyl(15), and (4-RO-1H-pyrazolo[3,4-b]pyridin-5yl)(phenyl-R)methanone where R was n-butyl and phenyl-R was 2,6-difluorophenyl, 2,6difluoro-4-methyphenyl, 2,4,6-trimethylphenyl, 2-fluorophenyl and 2-methylphenyl.[START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: causes of resistance[END_REF]

Figure 3 .

 3 Figure 3. CDK inhibitors, (4-RO-1H-pyrazolo[3,4-b]pyridin-5-yl)(phenyl)methanone showing alkoxy (R = n-butyl) substituents, and (4-butoxy-1H-pyrazolo[3,4-b]pyridin-5yl)(phenyl-R)methanone illustrated with 2,6-difluoro-4-methyl substituents on the phenyl-R ring.

Table 1 .

 1 The effect of statin properties on growth of pancreatic cancer cells and nude mice survival times

	Statin Properties	CAPAN-2 24 hours	CAPAN-2 48 hours	MiaPaCa-2 24 hours	MiaPaCa-2 48 hours	BxPc-3 24 hours	BxPc-3 48 hours	BxPc-3 72 hours	Mice Survival Times
	Desolvation	-1.3(0.5) -1.6(0.8) -1.4(0.3) -1.0(0.2) -1.3(0.8) -1.4(0.4) -0.7(0.6) -3.5(1.8)
	ΔG desolvation
	Lipophilicity	-0.3(0.3) 0.3(0.4)	-0.3(0.2) 0.2(0.1)	-0.3(0.4) 0(0.2)	0.5(0.3)	-1.1(1.0)
	ΔG lipophilicity
	Dipole	-1.0(1.2) 1.6(1.8)	-1.1(0.8) 1.3(0.5)	-1.3(1.9) 0(1.1)	2.4(1.5)	-5.6(4.4)
	Moment
	Molecular	-2.5(2.0) 0.2(3.0)	-2.7(1.4) 1.1(0.8)	-2.2(3.1) -1.1(1.7) 2.3(2.5)	-22.5(7.3)
	Volume*
	R 2	0.823	0.862	0.980	0.927	0.697	0.911	0.831	0.896
	F significance	0.320	0.256	0.038	0.139	0.510	0.169	0.300	0.196

Footnotes: Statin properties as per the general QSAR equation IC 50 (or survival times) = ΔG desolvation + ΔG lipophilicity + Dipole Moment + Molecular Volume. IC 50 (μM) for pancreatic cancer cell lines treated with statins and mice survival times (days) taken from Gbelkova 2008. The nude mice were xenotransplanted subcutaneously with human pancreatic adenocarcinoma cell line CAPAN-2.

Table 2 . Solvation energies, lipophilicities, dipole moments, molecular volumes of statins

 2

	Statin	Desolvation	Lipo-	Dipole	Molecular
		Energy	philicity	Moment	Volume
		Water	kcal/mol	Water	Water
		kcal/mol		D	cm 3 /mol
	Rosuvastatin Anion 90 o	74.2	-37.3	38.6	331
	Fluvastatin Anion 90º	85.1	-37.7	41.4	283
	Atorvastatin 90º Anion	75.7	-40.7	35.0	374
	Cerivastatin 90º Anion	94.1	-39.0	46.6	278
	Pitavastatin 90º Anion	93.8	-40.5	45.7	299
	Lovastatin Anion	87.8	-34.8	21.7	335
	Pravastatin Anion	104.3	-40.8	37.5	356
	Simvastatin Anion	84.5	-33.2	21.2	322
	Rosuvastatin Acid 90 o	32.7	-20.9	10.1	301
	Fluvastatin Acid 90º	23.7	-16.0	8.6	308
	Atorvastatin Acid 90º	26.7	-21.6	8.5	439
	Cerivastatin 90º Acid	37.2	-17.8	11.5	425
	Pitavastatin Acid 90º	26.4	-18.1	6.2	335
	Lovastatin Acid	25.5	-15.0	5.1	340
	Pravastatin Acid	35.3	-17.0	9.4	298
	Simvastatin Acid	25.4	-14.6	5.1	378
	Rosuvastatin Lactone 90 o	25.7	-16.8	7.7	316
	Fluvastatin Lactone 90 o	14.2	-16.4	1.5	304
	Atorvastatin Lactone 90 o	26.7	-23.3	5.5	355
	Cerivastatin Lactone 90 o	16.4	-15.9	10.1	342
	Pitavastatin Lactone 90 o	19.0	-17.5	9.1	300
	Lovastatin Lactone	18.1	-14.5	4.0	292
	Pravastatin Lactone	24.2	-15.6	4.8	295
	Simvastatin Lactone	20.7	-16.3	4.2	349


Nadjib Achir
email: nadjib.achir@inria.fr

Younes Bouchaala
email: younes.bouchaala@inria.fr

Paul Muhlethaler
email: paul.muhlethaler@inria.fr

Oyunchimeg Shagdar
email: oyunchimeg.shagdar@inria.fr

Eva Team

Comparison of Spatial Aloha and CSMA using Simple Stochastic Geometry Models for 1D and 2D Networks

Keywords: CSMA, slotted and non-slotted Aloha, spatial performance, stochastic geometry

Spatial throughput (i.e. throughput with spatial reuse) is important with new types of networks such as vehicular, sensor and military networks. The aim of this study is to compute the spatial throughput of Aloha and CSMA using tools for stochastic geometry. Our network nodes will be modeled as elements of a Poisson Point Process (PPP) of a one-or twodimensional space. Spatial Aloha can be modeled easily, the transmitting nodes are just selected with a given transmission probability. In spatial CSMA the nodes with the smallest backoff counter in their neighborhood will be selected to transmit and thus we can use random marks to perform the selection. We use the two models we have built to compare the spatial density of successful transmissions of CSMA and Aloha. To carry out a fair comparison, we will optimize both schemes by adjusting their parameters. For spatial Aloha, we can adapt the transmission probability, whereas for spatial CSMA we have to find the suitable carrier sense threshold.

The results obtained show that CSMA, when optimized, outperforms Aloha for nearly all the parameters of the network model values and we evaluate the gain of CSMA over Aloha. We also find interesting results concerning the effect of the model parameters on the performance of both Aloha and CSMA. The closed formulas we have obtained provide immediate evaluation of performance, whereas simulations may take minutes to give their results. Even if Aloha and CSMA are old protocols, this comparison of spatial performance is new and provides interesting and useful results.

I. INTRODUCTION

From the 60s to the 80s, most performance evaluations were performed on wired networks. At the end the twentieth century came the revolution of wireless networks and, in the domain of local area networks, the IEEE 802.11 standard also called WiFi. The dominant architecture of WiFi networks involves an access point. This node, which is generally connected to the Internet, exchanges packets with surrounding nodes which are linked to the access point. In each cell around an access point there is only one packet sent at each instant since all the nodes are within carrier sense range of the each other. More recently, great progress in wireless transmission technology has paved the way to much bigger networks with more massive transmission patterns. This phenomenon is known as spatial reuse. In VANETs, extending the networks along the roads leads to vast networks where spatial reuse must inevitably be present. Moreover, the density of the vehicles may vary greatly, which leads to further complications. The high density of communicating vehicles on a road using the IEEE 802.11p -a CSMA-based protocol -justifies the optimization of CSMA in networks with spatial reuse.

However, the access schemes used in these recent wireless networks also use the well-known Carrier Sense Multiple Access techniques (CSMA) as did the first Wireless LANs (WLANs) such as the IEEE 802.11 standard 1 . Therefore, a deeper understanding of CSMA with spatial reuse is needed. Aloha [START_REF] Abramson | The aloha system: Another alternative for computer communications[END_REF] was introduced in the early 70s. This is the simplest medium access control scheme since each node in the network simply transmits its packet at random and re-transmits it if necessary. Aloha can also, like CSMA, be used in these more recent networks which use the spatial reuse capability of the radio medium.

The remainder of this paper is organized as follows. Section II briefly describes related work. Section III describes the model proposed to study spatial Aloha and CSMA and develops the corresponding analytical model. The results of the model evaluating the influence of the parameters are reported in Section IV. Finally Section V concludes the paper.

II. RELATED WORK

The first studies on Aloha and CSMA were carried out in the 1970s resulting in several seminal studies such as the papers by Abramson [START_REF] Abramson | The aloha system: Another alternative for computer communications[END_REF] and Kleinrock [START_REF] Kleinrock | Packet switching in radio channels: Part I-carrier sense multiple-access modes and their throughput-delay characteristics[END_REF]. These two papers, and a great number of others using the same analytical model framework, assume that the shared radio medium can carry only one packet at a given time. This is a fundamental characteristic of these analyses. We will see below that a period of ten yearrs was necessary before papers investigating models with concurrent transmission on the medium appeared.

The study by Kleinrock [START_REF] Kleinrock | Packet switching in radio channels: Part I-carrier sense multiple-access modes and their throughput-delay characteristics[END_REF] was precise in its analysis of the carrier-sense effect but, the model of the back-off technique remained at a global level. It was only in 2000 that the paper by Bianchi [3] which proposed a Markov model for the transition between the back-off states of nodes in IEEE 802.11 networks made a significant step forward in the modeling of the CSMA back-off technique. However [START_REF] Bianchi | Performance Analysis of the IEEE 802.11 Distributed Coordination Function[END_REF] still did not introduce concurrent transmissions.

The study of Aloha has also received interesting contributions such as the analysis of slotted Aloha and the proposal of a scheme to stabilize Aloha [START_REF] Rivest | Network control by bayesian brodcast[END_REF]. However these improvements still do not consider concurrent transmissions.

Initial studies dealing with spatial reuse in spatial networks appeared in articles devoted to Aloha such as [START_REF] Ghez | Stability properties of slotted Aloha with multipacket reception capability[END_REF] and [START_REF] Baccelli | An aloha protocol for multihop mobile wireless networks[END_REF] and a model for slotted Aloha [START_REF] Ghez | Stability properties of slotted Aloha with multipacket reception capability[END_REF] was introduced by Ghez, Verdu and Schwartz in 1988. Their model included multipacket reception capability and was, to our best knowledge, the first quantitative model of wireless networks with spatial reuse. In [START_REF] Baccelli | An aloha protocol for multihop mobile wireless networks[END_REF] Baccelli et al. revisited the model and demonstrated that the probability of successful transmission in an Aloha network with spatial reuse could be accurately computed if the distance between the transmitter and the receiver was known. If this distance is known, the density of successful transmissions can also be computed.

In Aloha networks the randomization of the transmitters is stateless and the pattern of the simultaneously transmitting nodes is simple to evaluate. This explains the relative simplicity of the evaluation of Aloha. Describing the set of simultaneous transmission is a far more complex task for CSMA than for Aloha. Although in [START_REF] Jacquet | Mean number of transmissions with csma in a linear network[END_REF] the mean number of transmissions with CSMA is computed in a random linear network of vehicles, the authors only take into account the nearest interferer. In contrast, the present study takes the entire interference into account. The pattern of simultaneous transmissions in CSMA was evaluated in [START_REF] Muhlethaler | Throughput optimization in multihop csma mobile ad hoc networks[END_REF] using the Matern selection process [START_REF] Stoyan | Stochastic geometry and its applications[END_REF]. The authors in [START_REF] Busson | Point processes for interference modeling in csma/ca ad-hoc networks[END_REF] employ a similar process to evaluate interference but fail to compute the network throughput which is the corner stone of this paper. The model which was presented in [START_REF] Muhlethaler | Throughput optimization in multihop csma mobile ad hoc networks[END_REF], and improved in [START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF], forms the basis of the model that we use and extend in this paper. To the authors' best knowledge, there is no recent work pursuing these studies.

Concerning the comparison of spatial CSMA and spatial Aloha, this paper is apparently among the very few papers to tackle this subject. [START_REF] Blaszczyszyn | Comparison of the maximal spatial throughput of Aloha and CSMA in Wireless multihop Ad-Hoc Networks[END_REF] uses extensive simulations to evaluate performance whereas [START_REF] Kaynia | Performance of aloha and csma in spatially distributed wireless networks[END_REF] focuses on the outage probability and does not compute the spatial throughput. [START_REF] Manikandan | Throughput performance of aloha and csma mac protocol in power for wireless ad hoc networks[END_REF] focuses on a novel power control protocol to improve the aggregate throughput of the network. In addition to using of stochastic geometry models, the present paper has the originality of focusing its evaluation on transmission over a distance which is the mean distance between a node and its closest neighbor.

III. SYSTEM MODEL

In our model, the network nodes are randomly distributed according to a Poisson Point Process Φ. We denote by λ the intensity of the process. In our study we consider a 2D infinite plane, S = R2 or a 1D infinite line, S = R. The 2D model will be suitable for Mobile Ad-hoc NETworks (MANETs) and Wireless Sensor Networks (WSNs). The 1D model is more relevant for vehicular Ad-hoc NETworks (VANETs). We assume that the signal received in a transmission is the result of the combination of a random fading F and a power-law in the distance decay 1/r β where β is the decay factor and is generally between 2 and 6. In this paper the fading will be Rayleigh i.e exponentially distributed with parameter µ. Thus the signal received when the transmitter and the receiver are at distance r from each-other is F/l(r) 2 with l(r) = r β .

We use the well-accepted SINR (Signal over Interference and Noise Ratio) with a capture threshold T. In the following, we assume that there is no thermal noise (i.e. W ≡ 0). Dealing with thermal noise would be easy but is not done in this paper due to limitations of space. An even more realistic model than the SINR based on a graded SINR model using Shannon's law is possible in our framework though with an increased computational cost. This is left for further work because we consider that the SINR model is already a meaningful model. This study assumes that each node in Φ has a receiver at distance r from the transmitter. This distance will a typical distance of the PPP computed as the average distance to the closest neighbor 3 . We recall that this distance is r = 1/λ in 1D network and r = 1/ √ λ in 2D networks, see Figure 1.

In this paper we adopt a model where the network nodes are saturated. This means that at each instant each node always has a pending packet.

X i X k. X n X m X o X j. 1/ √ λ 1/ √ λ 1/ √ λ 1/ √ λ 1/ √ λ 1/ √ λ 1/ √ λ X l
Fig. 1. Transmission to a typical node in a 2D network over a distance which is the average distance to the closest neighbor. .

A. Analysis of spatial Aloha

The analysis of spatial Aloha (slotted and non-slotted) comes respectively from [START_REF] Baccelli | An aloha protocol for multihop mobile wireless networks[END_REF] and [START_REF] Blaszczyszyn | Interference and SINR coverage in spatial non-slotted aloha networks[END_REF]. In addition to the general network model described above, we need to add the nodes' transmission probability p. In slotted Aloha, p denotes the node's transmission probability at each slot. In non-slotted Aloha and in a given node, the duration between the end of a packet transmission and the beginning of the next transmission is exponentially distributed with rate . We can unify the notation with p of the slotted model if we set p = 1/(B + 1 )

where B denotes the duration of a packet 4 , see [START_REF] Blaszczyszyn | Interference and SINR coverage in spatial non-slotted aloha networks[END_REF]. For nonslotted Aloha we use the rain model introduced in [START_REF] Blaszczyszyn | Interference and SINR coverage in spatial non-slotted aloha networks[END_REF]. This model is shown to provide a very good estimation even when the nodes are fixed, as in the present model. For a complete presentation of the model for non-slotted Aloha the reader can refer to [START_REF] Blaszczyszyn | Interference and SINR coverage in spatial non-slotted aloha networks[END_REF].

With Rayleigh fading, the capture probability is directly linked to the Laplace of the interference and of the noise, we have p c (r, λ) = L I (µT l(r))L W (µT l(r)) = L I (µT l(r))(see [START_REF] Baccelli | An aloha protocol for multihop mobile wireless networks[END_REF]), with

L I (s) = exp -λp S (1 -L F (s/l(|x|)))dx
and L F (.) is the Laplace transform of the fading.

For the 2D and slotted Aloha case we have:

p c (r, λ) = exp -2π 2 λpr 2 T 2/β β sin(2π/β) , (1)
and for non-slotted Aloha:

p c (r, λ) = exp -4π 2 λpr 2 T 2/β (β + 2) sin(2π/β) . (2)
For the 1D and slotted Aloha case we have:

p c (r, λ) = exp -2πλprT 1/β β sin(π/β) , (3)
and for non-slotted Aloha:

p c (r, λ) = exp -4πλprT 1/β (β + 2) sin(π/β) . ( 4
)
The density of successful transmissions is:

D = λpp c (r, λ).
The maximization of D with respect to p is straightforward.

The table below provides the maximization (in p) of D for slotted Aloha and non-slotted Aloha for 2D networks.

2D

p opt D opt slotted Aloha

β sin(2π/β) 2π 2 λr 2 T 2/β β sin(2π/β)e -1 2π 2 r 2 T 2/β non-slotted Aloha sin(2π/β)(β+2) 4π 2 λr 2 T 2/β sin(2π/β)(β+2)e -1 4π 2 r 2 T 2/β
The case of 1D networks is given in the following table :

1D p opt D opt slotted Aloha β sin(π/β) 2πλrT 1/β β sin(π/β)e -1 2πrT 1/β non-slotted Aloha sin(π/β)(β+2) 4πλrT 1/β sin(π/β)(β+2)e -1 4πrT 1/β
The computations in the above tables assume that values of p opt are smaller than or equal to 1. This will be the case for the values of the system used in Section IV. If it is not the case, then p opt = 1 and D opt is obtained using the first four formulas of the above section setting p = 1 in p c (r, λ).

m i , X i m k , X k. random mark of a point in the PPP m n ,X n m o ,X o m p ,X p m l , X l m q ,X q m j ,X j.
elimination Fig. 2. Matern CSMA selection process and an example of over-elimination.

B. Analysis of spatial CSMA

To mimic the CSMA selection process, we use a Matern process in a marked PPP: (X i ∈ Φ,m i ), as is done in [START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF]. The m i are thus random numbers in [0, 1]. We denote by F i,j the fading for the transmission between X i and X j The Matern selection process elects the the points X k with the smallest random marks m k in their neighborhood. We need to introduce the carrier sense threshold P cs to precisely define the neighborhood of a point X i . We define

V(X i ) = {X j ∈ X i F i,j /l(|X i -X j |) > P cs } the neighborhood of X i . A node X i will be selected in the Matern selection process if and only if ∀X j ∈ V(X i ) m i < m j .
As can seen in Figure 2, X i has the smallest mark m i in its neighborhood and thus it will be selected in the Matern selection process. We observe that X q does in fact have a smaller mark, but it is not within node i's neighborhood. For the sake of simplicity, we have not taken into account any Rayleigh fading (F ≡ 1) in Figure 2 and thus the neighborhoods of the nodes are discs of the same radius. We can observe that the Matern selection process results in an over-elimination of nodes. When one node is eliminated by a node with a smaller mark, the node which has been eliminated should not continue to eliminate other nodes. But this overelimination can occur, as is shown in Figure 2 where node o is eliminated by node i, but node o goes on to eliminate node p in the Matern selection process, whereas in a real CSMA system, node o is correctly eliminated by node i, but, being eliminated, node o can not eliminate another node. We do not take this case into account in our model.

We note the medium access indicator of node

X i e i = 1I(∀X j ∈ V(X i ) m i < m j )
Proposition 1. The mean number of neighbors of a node is:

N = λ S P {F P cs l(|x|)}dx.
In a 2D network we have :

N = 2πλΓ(2/β) β(P cs µ) 2/β . ( 5
)
4 1 is the mean duration between the end of a packet and the start time of the next packet.

In a 1D network we have :

N = λΓ(1/β) β(P cs µ) 1/β . ( 6
)
This result is very simple. Let F 0 j be the fading between the node at the origin X i and node X j This is just the application of Slivnyak's theorem and Campbell's formula, see [START_REF] Baccelli | Stochastic Geometry and Wireless Networks, Volume I -Theory[END_REF], [START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF]

N = E 0 Xj ∈φ 1I(F 0 j l(|X j -X i |) P cs = λ S P {F P cs l(|x|)}dx
A straightforward computation provides the explicit value of N in 1D and 2D cases.

Proposition 2. The probability 5 p that a given node X 0 transmits i.e. e 0 = 1 is:

p = E 0 [e 0 ] = 1 -e -N N . (7)
Proof: The proof is obtained by computing the probability that a given node at the origin with a mark m = t is allowed to transmit. The result is then obtained by deconditioning on t. The details of the proof can be found in [START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF].

Proposition 3. The probability that X 0 transmits given that there is another node

X j ∈ Φ at distance r is p r with p r = p -e -Pcsµl(r) 1 -e -N N 2 - e -N N . (8)
Proof: The proof is the same as that of Proposition 2.

Proposition 4. Let us suppose that X 1 and X 2 are two points in Φ such that |X 1 -X 2 | = r. We suppose that node X 2 is retained by the selection process. The probability that X 1 is also retained is:

h(r) = 2 b(r)-N ( 1-e -N N -1-e -b(r) b(r) )(1 -e -Pcsµl(r) ) 1-e -N N -e -Pcsµl(r) 1-e -N N 2 -e -N N (9)
In a 2D network, we have:

b(r) = 2N -λ ∞ 0 2π 0 e -Pcsµ(l(τ )+l( √ τ 2 +r 2 -2rτ cos(θ))) dτ dθ.
(11) In a 1D network, we have:

b(r) = 2N -λ ∞ -∞
e -Pcsµ(l(τ )+l(|r-τ |)) dτ.

(12) 5 We use p here, the same notation as in Aloha since the two parameters have the same meaning. In Aloha p is chosen by the user and is the transmission probability. In CSMA p is the probability that a node transmits, which is governed by the carrier sense mechanism and thus p is a function of Pcs .

Proof: The proof can be found in [START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF].

Proposition 5. The probability of successfully receiving a packet at distance r in a CSMA system (carrier sense threshold P cs modeled by a Matern selection process) and with a capture threshold T is:

p c (r, P cs ) exp -λ S h(|x|) 1 + l(|x-r|) T l(r) dx . (13)
In a 2D network, we have:

p c (r, P cs ) exp -λ ∞ 0 2π 0 τ h(τ ) 1 + l( √ τ 2 +r 2 -2rτ cos(θ)) T l(r)
dτ dθ (14) In a 1D network, we have:

p c (r, P cs ) exp -λ ∞ -∞ h(τ ) 1 + l(|r-τ |) T l(r) dτ (15)
Proof: The idea of the proof is to consider a transmitter at the origin and to compute the probability of successful reception by a receiver at distance r. We assume the presence of another transmitting node at distance τ . According to proposition 4, the density of such nodes is λh(τ ). We approximate the interference by the interference of a Poisson Process of density λh(τ ). The result is obtained by integrating on τ . The details of the proof can be found in [START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF]. Proposition 6. The spatial density of successful transmissions is thus:

D = λpp c (r, P cs ). ( 16
)
This spatial density has a 1D and a 2D version and the values of p and p c (r, P cs ) are chosen accordingly. Proof: Proposition 6 is just the exploitation of propositions 2 and 5.

IV. RESULTS OF THE MODEL

In this section, we use the model to analyze the network performance and the influence of the model's parameters. We study the transmissions for pairs of source-destination nodes at distance r. r is set at 1/ √ λ or 1/λ for 2D and 1D networks respectively. r can be seen as a typical distance in these networks since it is the average distance between a node and its closest neighbor. Thus the transmitters are in the Poisson Point Process and for each transmitter, we create a random receiver at distance r.

A. Optimizing the density of successful transmissions with p for Aloha and the carrier sense threshold P cs for CSMA In this section we fix the parameters of the model λ, T and µ.

Figure 3 illustrates the density of successful transmissions for both slotted and non-slotted Aloha. This density starts to increase, reaches a maximum for a value of the transmission probability p opt computed in Section III-A, and then decreases.

For CSMA we vary P cs to maximize the density of successful transmissions. The probability of transmission p in CSMA increases when P cs increases, which is because with larger values of P cs transmission is easier. But, in contrast, when P cs increases, p c (r, P cs ) decreases, as can be shown using the equation of proposition 4. When P cs increases, h(τ ) increases and thus p c (r, P cs ) decreases. Thus there is an optimal value of P cs which optimizes the density of successful transmissions. Studying some significant examples, we have seen that the density of successful transmissions always has the same behavior, as shown in Figure 3. For small values of P cs and when we increase P cs , p increases faster than p c (r, P cs ) decreases, and thus the density of successful transmissions is an increasing function of P cs . This density reaches a maximum for a given value of P cs and then becomes a decreasing function of P cs . We assume that this is always the case although it seems difficult to prove it in the equations. We use Maple to numerically compute this optimum of the density of successful transmissions.

B. Effect of the fading rate µ

For Aloha, computing the probability of successful transmissions and the density of successful transmissions shows that the fading rate µ disappears in the equations 6 . Thus this parameter has no effect on the performance of Aloha (slotted and non-slotted). For CSMA we observe that in the probability of transmission p found in Proposition 2, we can isolate µP cs . It is the same for p c (r, P cs ). Thus if we multiply µ by x, we can obtain the same performance by dividing P cs by x. Thus µ does not influence the global performance of the system; the optimum density of successful transmissions, the probability of capture p c (r, P cs ) and the probability of transmission p at the optimum value of P cs .

C. Effect of the density of nodes λ

For both slotted and non-slotted Aloha, the maximum density of successful transmissions is obtained by we exploiting formulas of Section III-A.

We study transmissions over the average distance to the closest neighbor, thus λr = 1 for 1D networks and λr 2 = 1 for 2D networks. We use the following parameters T = 1, µ = 10 and β = 4. In the above formula, we observe that the maximum density of successful transmissions is linear in λ for 1D and 2D networks, as is also shown in Figures 4 and5.

For CSMA, we compute the optimum density of successful transmissions when P cs is optimized versus λ the density of nodes in the network. The results of these computations are shown in Figure 4 for 2D networks and in Figure 5 for 1D networks. Our numerical study shows that the density of successful transmissions is linear in λ. This means that the maximum of the product of pp c (r, P cs ) does not depend on Figures 4 and 5 also show the density of successful transmissions for CSMA when the carrier sense threshold is constant and taken as the optimal value for λ = 1. The loss is already significant for small values of λ: 26% for λ = 0.1 and is even more significant for large values of λ : 80% for λ = 10 in 2D networks and 85% for λ = 10 in 1D networks. For instance, this means that, in a VANET, the channel cannot be used efficiently if the carrier sense threshold is not properly optimized according to the density of vehicles. When λ = 0.1 and if we use the optimization for λ = 1 we do not have any restriction on the transmission rights but rather an excess of transmission rights. The problem arises from the probability of success for a given transmission. When λ = 10 and if we use the optimization for λ = 1 we have a tight restriction on the transmission rights, whereas a given transmission is very well protected by the CSMA scheme and thus every transmission is nearly always successful. The model shows that the problem concerning the access right is much more detrimental to the global throughput than collisions would have been if the density of nodes had been overestimated.

We have studied the probability of capture when the throughput is optimized. We observed that, for CSMA, the optimum throughput is not obtained when most of the transmissions are successful but rather when the success rate is around 55% in 2D networks and around 70% in 1D networks. The numerical results we obtained show that, at the optimum, p c (r, P cs ) does not depend on λ and we also deduce that p does not depend on λ. This is an interesting result which is not brought to light using the analytical formulas. In Aloha the probability of capture when the throughput is optimized is 1/e as shown by the computations in Section III-A.

When we compare the density of successful transmissions for Aloha with the density of successful transmissions for CSMA, we find that the gain of CSMA is 57% and 78% with non-slotted and 18% and 33% with slotted Aloha for 2D and 1D networks respectively, see Figures 4 and5. We observe that this gain is particularly significant in 1D networks. However, when the carrier sense threshold is not optimized, Aloha can offer a larger density of successful transmissions, as shown in Figures 4 and5. A very useful property of Aloha is that when we consider transmissions over the average distance to the closest neighbor, the probability which optimizes the density of successful transmissions does not depend on λ (see the formulas for p opt in Section III-A 7 ). Thus in contrast to CSMA, there is no risk of an unsuitable tuning for Aloha. This represents a huge advantage of Aloha. In a VANET, where the density of vehicles may greatly vary, we may wonder whether Aloha would not be a better option than CSMA if we can not dynamically adjust the carrier threshold.

D. Effect of the capture threshold T

We study the effect of the capture threshold on the maximum density of successful transmissions. In Figure 6 and Figure 7 we plot the maximum density of successful transmissions for T varying from 0.01 to 10 respectively for 7 this is because in 1D networks λr = 1 and in 2D networks λr 2 = 1 2D and 1D networks. We observe that dividing the capture threshold by 100 leads to multiplying the density of successful transmissions by 5.6 and 1.9 for 2D and 1D CSMA networks.

For slotted and non-slotted Aloha, the density of successful transmissions is multiplied by 10 and 3.16 for 2D and 1D networks respectively. This means that a small capture threshold is much more beneficial in 2D networks. Our study of the analytical model does not show any obvious scaling of the density of successful transmissions with the capture threshold T in CSMA networks. For Aloha (slotted and non-slotted) the scaling is respectively 1/T 2/β and 1/T 1/β . For T = 10, CSMA outperforms slotted and non-slotted Aloha by 68% and 124% in 2D networks. For T = 10, CSMA outperforms slotted and non-slotted Aloha by 65% and 119% in 1D networks

E. Effect of the transmission decay β

In Figures 8 and9, we plot the maximum density of successful transmissions for β varying from 2 to 6 respectively for 2D and 1D networks.

In 2D CSMA networks, we observe that the maximum density of successful transmissions is multiplied by 1.91 when β varies from 2.5 to 6. For linear (1D) CSMA networks the maximum density of successful transmissions is multiplied by 1.32 when β varies from 2 to 6. As for the capture threshold, the effect of a large transmission decay is less beneficial for 1D networks than for 2D networks. Our study does not show any apparent scaling of the density of successful transmissions with the capture threshold β.

In 2D Aloha networks, we observe that the maximum density of successful transmissions is multiplied by 3.54 and 2.62 for slotted and non-slotted Aloha when β varies from 2.5 to 6. In 1D Aloha networks, the maximum density of successful transmissions is multiplied by 1.5 and 1.0 for slotted and non-slotted Aloha when β varies from 2 to 6. The scaling of Aloha's performance with β is given by the formulas in Section III-A. V. CONCLUSION In this paper, we present spatial models for Aloha and CSMA networks. These models providing closed formulas result in instantaneous performance evaluation of the two schemes. We show the importance of optimizing tuning parameters according to the network parameters. We have shown that the optimized density of successful transmissions (when transmissions over the average distance to the closest neighbor are considered) scales linearly with the density of the nodes, both for Aloha and CSMA networks. However, in CSMA the carrier sense threshold must be adapted to the density of the nodes, whereas in Aloha a constant transmission probability can be used. In CSMA, using a constant carrier threshold leads to a very significant loss in the network's global throughput. This effect is much more detrimental when the density of the nodes in the network is underestimated than when it is overestimated.

For Aloha networks, when we study transmissions over the average distance to the closest neighbor, the optimization does not depend on the density of nodes, which is a very interesting property. Thus in Aloha networks, the density of successful transmissions easily scales linearly in λ when we vary λ whereas in CSMA networks the protocol must be carefully tuned to obtain this scaling. However, CSMA (when optimized) outperforms Aloha for nearly all values of the model's parameters. For instance, for λ = 1 and T = 1 the improvement brought about by CSMA compared to Aloha is notable. This improvement is very significant for λ = 1 and T = 10.

Our models allow the influence of the network model's parameters to be studied qualitatively. For instance, the improvement in the performance can be accurately computed when T is decreased or when β is increased. The models show that the influence of T and β on the performance is more pronounced for 2D than for 1D networks.

The models for Aloha have already been compared with simulation results. This comparison shows a very good match-ing between the model and the simulation results. This is because there is no approximation made in the Aloha model. In contrast, there are a number of approximations made in the CSMA model such as the selection process and the back-off scheme. However, initial simulations suggest that the CSMA model captures the essential behavior of the carrier sense technique and provides reasonably accurate results.

  with b(r) = 2N -λ S e -Pcsµ(l(|x|)+l(|r-x|) dx.

Fig. 3 .

 3 Fig. 3. Density of successful transmissions versus carrier threshold (T =1, µ = 10, β = 4).

Fig. 4 .Fig. 5 .

 45 Fig. 4. Density of successful transmissions versus density of nodes (T =1, µ = 10, β = 4). Spatial network (2 D)

Fig. 6 .Fig. 7 .

 67 Fig. 6. Density of successful transmissions versus capture threshold T for 2D networks (λ=1, µ = 10, β = 4).

Fig. 8 .Fig. 9 .

 89 Fig. 8. Density of successful transmissions versus the decay factor β for 2D networks (λ=1, µ = 10, T = 1).

CSMA is also used in IEEE 802.15.4, a widely used protocol for sensor networks.

The power received P = P 0 F l(r) and we set P 0 = 1

Dealing with a distance following the distribution of the n-th nearest neighbor [generalized gamma distribution] is possible in our frame but will incur more computation

This because the fading affects the transmission and the interference equally.λ. This is an interesting result and one which is not easily apparent in the analytical formulas of pp c (r, P cs ).