Evaluation of regional and distant metastases in head and neck squamous cell carcinoma patients with special reference to the assessment of regional lymph nodes in oral cancer

For optimal treatment planning, a thorough assessment of the metastatic status of mucosal squamous cell carcinoma of the head and neck (HNSCC) is required. Current imaging methods do not allow the recognition of all patients with metastatic disease. Therefore, elective treatment of the cervical lymph nodes is usually given to patients in whom the risk of subclinical metastasis is estimated to exceed 15-20%. The objective of this study was to improve the pre-treatment evaluation of patients diagnosed with HNSCC. Particularly, we aimed at improving the identification of patients who will benefit from elective neck treatment. Computed tomography (CT) of the chest and abdomen was performed prospectively for 100 patients diagnosed with HNSCC. The findings were analysed to clarify the indications for this examination in this patient group. CT of the chest influenced the treatment approach in 3% of patients, while CT of the abdomen did not reveal any significant findings. Our results suggest that CT of the chest and abdomen is not indicated routinely for patients with newly diagnosed HNSCC but can be considered in selected cases. Retrospective analysis of 80 patients treated for early stage squamous cell carcinoma of the oral tongue was performed to investigate the potential benefits of elective neck treatment and to examine whether histopathological features of the primary tumour could be used in the prediction of occult metastases, local recurrence, or/and poor survival. Patients who had received elective neck treatment had significantly fewer cervical recurrences during the follow-up when compared to those who only had close observation of the cervical lymph nodes. Elective neck treatment did not result in survival benefit, however. Of the histopathological parameters examined, depth of infiltration and pT-category (representing tumour diameter) predicted occult cervical metastasis, but only the pT-category predicted local recurrence. Depth of infiltration can be used in the identification of at risk patients but no clear cut-off value separating high-risk and low-risk patients was found. None of the histopathological parameters examined predicted survival. Sentinel lymph node (SLN) biopsy was studied as a means of diagnosing patients with subclinical cervical metastases. SLN biopsy was applied to 46 patients who underwent elective neck dissection for oral squamous cell carcinoma. In addition, SLN biopsy was applied to 13 patients with small oral cavity tumours who were not intended to undergo elective neck dissection because of low risk of occult metastasis. The sensitivity of SLN biopsy for finding subclinical cervical metastases was found to be 67%, when SLN status was compared to the metastatic status of the rest of the neck dissection specimen. Of the patients not planned to have elective neck dissection, SLN biopsy revealed cervical metastasis in 15% of the patients. Our results suggest that SLN biopsy can not yet entirely replace elective neck dissection in the treatment of oral cancer, but it seems beneficial for patients with low risk of metastasis who are not intended for elective neck treatment according to current treatment protocols.

Cooperation of Notch and Ras/MAPK signaling pathways in human breast carcinogenesis

Background: Recent studies have implicated aberrant Notch signaling in breast cancers. Yet, relatively little is known about the pattern of expression of various components of the Notch pathway, or its mechanism of action. To better understand the role of the Notch pathway in breast cancer, we have undertaken a detailed expression analysis of various Notch receptors, their ligands, and downstream targets at different stages of breast cancer progression. Results: We report here that there is a general increase in the expression levels of Notch 1, 2, 4, Jagged1, Jagged2, and Delta-like 4 proteins in breast cancers, with simultaneous upregulation of multiple Notch receptors and ligands in a given cancer tissue. While Notch3 and Delta-like1 were undetectable in normal tissues, moderate to high expression was detected in several cancers. We detected the presence of active, cleaved Notch1, along with downstream targets of the Notch pathway, Hes1/Hes5, in similar to 75% of breast cancers, clearly indicating that in a large proportion of breast cancers Notch signaling is aberrantly activated. Furthermore, we detected cleaved Notch1 and Hes1/5 in early precursors of breast cancers - hyperplasia and ductal carcinoma in situ suggesting that aberrant Notch activation may be an early event in breast cancer progression. Mechanistically, while constitutively active Notch1 alone failed to transform immortalized breast cells, it synergized with the Ras/MAPK pathway to mediate transformation. This cooperation is reflected in vivo, as a subset of cleaved Notch positive tumors additionally expressed phopsho-Erk1/2 in the nuclei. Such cases exhibited high node positivity, suggesting that Notch-Ras cooperation may lead to poor prognosis. Conclusions: High level expression of Notch receptors and ligands, and its increased activation in several breast cancers and early precursors, places Notch signaling as a key player in breast cancer pathogenesis. Its cooperation with the Ras/MAPK pathway in transformation offers combined inhibition of the two pathways as a new modality for breast cancer treatment.

Immediate Breast Reconstructions in the Treatment of Breast Cancer

Breast cancer is the most common form of potentially fatal cancer in women in the Western world. Better understanding of the breast cancer disease process together with developments in treatments have led to improved survival and reduced risk of recurrence, significantly influencing the acceptance of breast reconstructions as part of breast cancer treatment. Skin-sparing mastectomy followed by immediate breast reconstruction has proved superior to other forms of breast reconstruction in terms of aesthetic outcome. However, due to the relatively recent introduction of skin-sparing mastectomy concerns on the surgical and oncological safety of the operation persist. The aim of the present study is to evaluate the surgical and oncological safety of skin-sparing mastectomy and immediate breast reconstruction in a consecutive patient series with ensuing follow-up. Subsequent aims of the study are to examine possibilities of reducing surgical complications of the operation and to assess the feasibility of sentinel node biopsy together with immediate breast reconstruction. The study population comprises a consecutive series of patients having undergone skin-sparing mastectomy followed by immediate breast reconstruction at the Helsinki University Central Hospital between 1992 and 2006. In Study I, the hospital records of 207 patients, operated between 1992 and 2001, were analyzed for surgical complications and recurrences of breast cancer during follow-up. In Study II, 60 consecutive patients were randomized into either conventional diathermy or radiofrequency coagulation groups to examine possibilities of reducing skin-flap complications. Study III consists of 62 consecutive breast cancer patients evaluated for the feasibility of sentinel node biopsy simultaneously with immediate breast reconstruction. In Study IV, hospital records were analyzed to examine local recurrence of breast cancer in a consecutive series of 146 patients with Stage I or II disease. Post-operative complications in Study I included native skin-flap necrosis (10.1%), hematoma (10.1%), anastomose thrombosis (5.3%), infection (3.4%), hernia (2.6%) and loss of one microvascular flap (0.7%). The Stage I and II patients in Study IV had a local recurrence rate of 2.7%, an isolated regional lymph node recurrence rate of 2.1% and a systemic recurrence rate of 2.7%, during a mean follow-up time of 51 months. The Stage III patients in study I had a locoregional recurrence rate of 31.3% during follow-up. Radiofrequency coagulation in Study II did not decrease skin-flap complications when compared with conventional diathermy. An increased skin-flap complication rate in Study II was associated with smoking and the type of skin incision used. In Study III, eleven patients had tumor positive sentinel nodes, nine of which were detected intraoperatively. Skin-sparing mastectomy followed by immediate breast reconstruction is a safe procedure both surgically and oncologically, especially for early stage breast cancer. Tennis racket type incision is associated with an increased skin-flap complication rate. Sentinel node biopsy with intraoperative assessment of sentinel node metastases is feasible in patients undergoing immediate breast reconstruction.

Deregulation of miR-126 expression in colorectal cancer pathogenesis and its clinical significance

In this study, we investigated the expression profiles and clinicopathological significance of miR-126 in large cohort of patients with colorectal cancers as well the cellular repercussions of miR-126 in colon cancer cells along with its targets in-vitro. Down regulation of miR-126 expression was associated with histological subtypes, peri-neural tumour infiltration, microsatellite instability and pathological staging of colorectal cancers (p<0.05). Low miR-126 expression was also associated with poorer survival in patients with colorectal cancer. Analysis of matched tissues from the same patient revealed that approximately 70% of the tested patients had similar levels of expression of miR-126 in primary cancer and cancer metastases in both lymph node and distant metastases. In addition, induced overexpression of miR-126 showed reduced cell proliferation, increased apoptosis and decreased accumulation of cells in the G0-G1 phase of the colon cancer cells. Furthermore, SW480(+miR-126) cells showed reduced BCL-2 and increased P53 protein expression. To conclude, deregulation of miR-126 in colorectal cancer at the tissue and cellular levels as well as its correlation with various clinicopathological parameters confirm the cancer suppressive role of miR-126 in colorectal cancer.

Delay Optimal Scheduling in a Two-Hop Vehicular Relay Network

We study a scheduling problem in a wireless network where vehicles are used as store-and-forward relays, a situation that might arise, for example, in practical rural communication networks. A fixed source node wants to transfer a file to a fixed destination node, located beyond its communication range. In the absence of any infrastructure connecting the two nodes, we consider the possibility of communication using vehicles passing by. Vehicles arrive at the source node at renewal instants and are known to travel towards the destination node with average speed v sampled from a given probability distribution. Th source node communicates data packets (or fragments) of the file to the destination node using these vehicles as relays. We assume that the vehicles communicate with the source node and the destination node only, and hence, every packet communication involves two hops. In this setup, we study the source node's sequential decision problem of transferring packets of the file to vehicles as they pass by, with the objective of minimizing delay in the network. We study both the finite file size case and the infinite file size case. In the finite file size case, we aim to minimize the expected file transfer delay, i.e. expected value of the maximum of the packet sojourn times. In the infinite file size case, we study the average packet delay minimization problem as well as the optimal tradeoff achievable between the average queueing delay at the source node buffer and the average transit delay in the relay vehicle.

Tumour markers as prognostic factors of survival of gastric cancer patients

The incidence of gastric cancer in the last decades has declined rapidly in the industrialised countries. Worldwide, however, gastric cancer is still the second most common cause of cancer death. Although surgery is currently the most effective treatment, the rapid progress in adjuvant chemotherapy and radiation therapy requires a re-evaluation of prognosis assessment. The TNM staging system of the UICC is ubiquitously used; it groups patients by decreasing survival times from stage I to stage IV based on the spread of disease, i.e. depth of tumour penetration (T), extent of spread to lymph nodes (N), and the presence or absence of distant (M) metastases. This is by far the most consistent prognostic classification system today. However, even within the stage groups there are patients that follow a varying course of disease. Our knowledge of the molecular differences between tumours of the same stage and morphology has been accumulating over the years and methods for a more accurate assessment of the phenotype of neoplasias are of value when evaluating the prognosis of individual patients with gastric cancer. In this study, the immunohistochemical expression of tumour markers involved in different phases in tumourigenesis was examined. The aim was to find new markers which could provide prognostic information in addition to what is provided by the TNM variables. A total of 337 specimens from the primary tumour of patients who underwent surgery for gastric cancer were collected and the immunohistochemical expression of seven different biomarkers was analysed. DNA ploidy and S-phase fraction (SPF) was assessed by flow cytometry. Finally, all biomarkers and clinicopathological prognostic factors were combined and evaluated by a multivariate Cox regression model to elucidate which specific factors provide independent prognostic information. By univariate survival analysis the following variables were significant prognostic factors: epithelial and stromal syndecan-1 expression, stromal tenascin-C expression, expression of tumour-associated trypsin inhibitor (TATI) in cancer cells, nuclear p53 expression, nuclear p21 expression, DNA ploidy, and SPF. By multivariate survival analysis adjusted for all available clinicopathological and biomolecular variables, p53 expression, p21 expression, and DNA ploidy emerged as independent prognostic biomarkers, together with penetration depth of the tumour, presence of nodal metastases, surgical cure of the cancer, and age of the patient at the time of diagnosis.

Matrix metalloproteinases as biomarkers in premalignant and malignant tumors of the human skin

The incidence of non-melanoma skin cancer is increasing worldwide. Basal cell carcinoma followed by squamous cell carcinoma and malignant melanoma are the most frequent skin tumors. Immunosuppressed patients have an increased risk of neoplasia, of which non-melanoma skin cancer is the most common. Matrix metalloproteinases (MMPs) are proteolytic enzymes that collectively are capable of degrading virtually all components of the extracellular matrix. MMPs can also process substrates distinct from extracellular matrix proteins and influence cell proliferation, differentiation, angiogenesis, and apoptosis. MMP activity is regulated by their natural inhibitors, tissue inhibitors of metallopro-teinases (TIMPs). In this study, the expression patterns of MMPs, TIMPs, and certain cancer-related molecules were investigated in premalignant and malignant lesions of the human skin. As methods were used immunohistochemisty, in situ hybridization, and reverse transcriptase polymerase chain reaction (RT-PCR) from the cell cultures. Our aim was to evaluate the expression pattern of MMPs in extramammary Paget's disease in order to find markers for more advanced tumors, as well as to shed light on the origin of this rare neoplasm. Novel MMPs -21, -26, and -28 were studied in melanoma cell culture, in primary cutaneous melanomas, and their sentinel nodes. The MMP expression profile in keratoacanthomas and well-differentiated squamous cell carcinomas was analyzed to find markers to differentiate benign keratinocyte hyperproliferation from malignantly transformed cells. Squamous cell carcinomas of immunosuppressed organ transplant recipients were compared to squamous cell carcinomas of matched immunocompetent controls to investigate the factors explaining their more aggressive nature. We found that MMP-7 and -19 proteins are abundant in extramammary Paget's disease and that their presence may predict an underlying adenocarcinoma in these patients. In melanomas, MMP-21 was upregulated in early phases of melanoma progression, but disappeared from the more aggressive tumors with lymph node metastases. The presence of MMP-13 in primary melanomas and lymph node metastases may relate to more aggressive disease. In keratoacanthomas, the expression of MMP-7 and -9 is rare and therefore should raise a suspicion of well-differentiated squamous cell carcinomas. Furthermore, MMP-19 and p16 were observed in benign keratinocyte hyperproliferation of keratoacanthomas, whereas they were generally lost from malignant keratinocytes of SCCs. MMP-26 staining was significantly stronger in squamous cell carcinomas and Bowen s disease samples of organ transplant recipients and it may contribute to the more aggressive nature of squamous cell carcinomas in immunosuppressed patients. In addition, the staining for MMP-9 was significantly stronger in macrophages surrounding the tumors of the immunocompetent group and in neutrophils of those patients on cyclosporin medication. In conclusion, based on our studies, MMP-7 and -19 might serve as biomarkers for more aggressive extramammary Paget's disease and MMP-21 for malignant transformation of melanocytes. MMP -7, -9, and -26, however, could play an important role in the pathobiology of keratinocyte derived malignancies.

A Slotted Aloha joint MAC-cum-Routing protocol for data gathering sensor network applications

In this paper we have proposed and implemented a joint Medium Access Control (MAC) -cum- Routing scheme for environment data gathering sensor networks. The design principle uses node 'battery lifetime' maximization to be traded against a network that is capable of tolerating: A known percentage of combined packet losses due to packet collisions, network synchronization mismatch and channel impairments Significant end-to-end delay of an order of few seconds We have achieved this with a loosely synchronized network of sensor nodes that implement Slotted-Aloha MAC state machine together with route information. The scheme has given encouraging results in terms of energy savings compared to other popular implementations. The overall packet loss is about 12%. The battery life time increase compared to B-MAC varies from a minimum of 30% to about 90% depending on the duty cycle.

An optimal weighted-average congestion based pricing scheme for enhanced QoS

Pricing is an effective tool to control congestion and achieve quality of service (QoS) provisioning for multiple differentiated levels of service. In this paper, we consider the problem of pricing for congestion control in the case of a network of nodes under a single service class and multiple queues, and present a multi-layered pricing scheme. We propose an algorithm for finding the optimal state dependent price levels for individual queues, at each node. The pricing policy used depends on a weighted average queue length at each node. This helps in reducing frequent price variations and is in the spirit of the random early detection (RED) mechanism used in TCP/IP networks. We observe in our numerical results a considerable improvement in performance using our scheme over that of a recently proposed related scheme in terms of both throughput and delay performance. In particular, our approach exhibits a throughput improvement in the range of 34 to 69 percent in all cases studied (over all routes) over the above scheme.

Conjunctival Melanoma in Finland 1967-2000: a population-based study

Eighty-five new cases of conjunctival melanoma (CM) were diagnosed in Finland between 1967 and 2000. The annual crude incidence of CM was 0.51 per million inhabitants. The average age-adjusted incidence of 0.54 doubled during the study period, analogous to the increase in the incidence of cutaneous malignant melanoma during this period, suggesting a possible role for ultraviolet radiation in its pathogenesis. Nonlimbal tumors were more likely than limbal ones to recur and they were associated with decreased survival. Increasing tumor thickness and recurrence of the primary tumor were other clinical factors related to death from CM. The histopathologic specimens of 85 patients with CM melanoma were studied for cell type, mitotic count, tumor-infiltrating lymphocytes and macrophages, mean vascular density, extravascular matrix loops and networks, and mean diameter of the ten largest nucleoli (MLN). The absence of epithelioid cells, increasing mitotic count and small MLN were associated with shorter time to recurrence according to the Cox univariate regression. None of the histopathologic variables was associated with mortality from CM. Four (5%) patients had a CM limited to the cornea without evidence of a tumor other than primary acquired melanosis of the conjunctiva. Because there are no melanocytes in the cornea, the origin of these melanomas most likely is the limbal conjunctiva. All four corneally displaced CM were limited to the epithelium, and none of the patients developed metastases. An anatomic sub-classification based on my patients and world literature was developed for corneally displaced CM. In 20 patients the metastatic pattern could be determined. Ten patients had initial systemic metastases detected, nine had initial regional metastases, and in one case the two types were detected simultaneously. The patients most likely to develop either type of initial metastases were those with nonlimbal conjunctival melanoma, those with a primary tumor more than 2 mm thick, and those with a recurrent conjunctival melanoma. Approximately two thirds of the patients had limbal CM, a location associated with good prognosis. One third, however, had a primary CM originating outside the limbus. In these patients the chance of developing local recurrences as well as systemic metastases was significantly higher than in patients with limbal CM. Each recurrence accompanies an increased risk of developing metastases, and recurrences contribute to death along with increasing tumor thickness and nonlimbal tumor location. In my data, an equal number of patients with initial locoregional and systemic metastasis existed. Patients with limbal primary tumors less than 2 mm in thickness rarely experienced metastases, unless the tumor recurred. Consequently, the patients most likely to benefit from sentinel lymph node biopsy are those who have nonlimbal tumors, CM that are over 2 mm thick, or recurrent CM. The histopathology of CM differs from that of uveal melanoma. Microvascular factors did not prove to be of prognostic importance, possibly due to the fact that CM at least as often disseminates first to the regional lymph nodes, unlike uveal melanoma that almost always disseminates hematogenously.

Prognostic markers in head and neck carcinoma

Head and neck squamous cell carcinoma (HNSCC) is the sixth most common cancer worldwide. Well-known risk factors include tobacco smoking and alcohol consumption. Overall survival has improved, but is still low especially in developing countries. One reason for this is the often advanced stage of the disease at the time of diagnosis, but also lack of reliable prognostic tools to enable individualized patient treatment to improve outcome. To date, the TNM classification still serves as the best disease evaluation criterion, although it does not take into account the molecular basis of the tumor. The need for surrogate molecular markers for more accurate disease prediction has increased research interests in this field. We investigated the prevalence, physical status, and viral load of human papillomavirus (HPV) in HNSCC to determine the impact of HPV on head and neck carcinogenesis. The prevalence and genotyping of HPV were assessed with an SPF10 PCR microtiter plate-based hybridization assay (DEIA), followed by a line probe-based genotyping assay. More than half of the patients had HPV DNA in their tumor specimens. Oncogenic HPV-16 was the most common type, and coinfections with other oncogenic and benign associated types also existed. HPV-16 viral load was unevenly distributed among different tumor sites; the tonsils harbored significantly greater amounts of virus than other sites. Episomal location of HPV-16 was associated with large tumors, and both integrated and mixed forms of viral DNA were detected. In this series, we could not show that the presence of HPV DNA correlated with survival. In addition, we investigated the prevalence and genotype of HPV in laryngeal carcinoma patients in a prospective Nordic multicenter study based on fresh-frozen laryngeal tumor samples to determine whether the tumors were HPV-associated. These patients were also examined and interviewed at diagnosis for known risk factors, such as tobacco smoking and alcohol consumption, and for several other habituations to elucidate their effects on patient survival. HPV analysis was performed with the same protocols as in the first study. Only 4% of the specimens harbored HPV DNA. Heavy drinking was associated with poor survival. Heavy drinking patients were also younger than nonheavy drinkers and had a more advanced stage of disease at diagnosis. Heavy drinkers had worse oral hygiene than nonheavy drinkers; however, poor oral hygiene did not have prognostic significance. History of chronic laryngitis, gastroesophageal reflux disease, and orogenital sex contacts were rare in this series. To clarify why vocal cord carcinomas seldom metastasize, we determined tumor lymph vessel (LVD) and blood vessel (BVD) densities in HNSCC patients. We used a novel lymphatic vessel endothelial marker (LYVE-1 antibody) to locate the lymphatic vessels in HNSCC samples and CD31 to detect the blood microvessels. We found carcinomas of the vocal cords to harbor less lymphatic and blood microvessels than carcinomas arising from sites other than vocal cords. The lymphatic and blood microvessel densities did not correlate with tumor size. High BVD was strongly correlated with high LVD. Neither BVD nor LVD showed any association with survival in our series. The immune system plays an important role in tumorigenesis, as neoplastic cells have to escape the cytotoxic lymphocytes in order to survive. Several candidate HLA class II alleles have been reported to be prognostic in cervical carcinomas, an epithelial malignancy resembling HNSCC. These alleles may have an impact on head and neck carcinomas as well. We determined HLA-DRB1* and -DQB1* alleles in HNSCC patients. Healthy organ donors served as controls. The Inno-LiPA reverse dot-blot kit was used to identify alleles in patient samples. No single haplotype was found to be predictive of either the risk for head and neck cancer, or the clinical course of the disease. However, alleles observed to be prognostic in cervical carcinomas showed a similar tendency in our series. DRB1*03 was associated with node-negative disease at diagnosis. DRB1*08 and DRB1*13 were associated with early-stage disease; DRB1*04 had a lower risk for tumor relapse; and DQB1*03 and DQB1*0502 were more frequent in controls than in patients. However, these associations reached only borderline significance in our HNSCC patients.

Computing TCP Throughput in a UMTS Network

We compute the throughput obtained by a TCP connection in a UMTS environment. For downloading data at a mobile terminal, the packets of each TCP connection are stored in separate queues at the base station (node B). Also due to fragmentation of the TCP packets into Protocol Data Units (PDU) and link layer retransmissions of PDUs there can be significant delays at the queue of the node B. In such a scenario the existing models of TCP may not be sufficient. Thus, we provide a new approximate TCP model and also obtain new closed-form expressions of mean window size. Using these we obtain the throughput of a TCP connection which matches with simulations quite well.

Correlation, Coding, and Cooperation in Wireless Sensor Networks

We consider a single-hop data-gathering sensor network, consisting of a set of sensor nodes that transmit data periodically to a base-station. We are interested in maximizing the lifetime of this network. With our definition of network lifetime and the assumption that the radio transmission energy consumption forms the most significant portion of the total energy consumption at a sensor node, we attempt to enhance the network lifetime by reducing the transmission energy budget of sensor nodes by exploiting three system-level opportunities. We pose the problem of maximizing lifetime as a max-min optimization problem subject to the constraint of successful data collection and limited energy supply at each node. This turns out to be an extremely difficult optimization to solve. To reduce the complexity of this problem, we allow the sensor nodes and the base-station to interactively communicate with each other and employ instantaneous decoding at the base-station. The chief contribution of the paper is to show that the computational complexity of our problem is determined by the complex interplay of various system-level opportunities and challenges.

DMT of Multi-hop Cooperative Networks-Part I: K-Parallel-Path Networks

We consider single-source, single-sink multi-hop relay networks, with slow-fading Rayleigh fading links and single-antenna relay nodes operating under the half-duplex constraint. While two hop relay networks have been studied in great detail in terms of the diversity-multiplexing tradeoff (DMT), few results are available for more general networks. In this two-part paper, we identify two families of networks that are multi-hop generalizations of the two hop network: K-Parallel-Path (KPP) networks and Layered networks. In the first part, we initially consider KPP networks, which can be viewed as the union of K node-disjoint parallel paths, each of length > 1. The results are then generalized to KPP(I) networks, which permit interference between paths and to KPP(D) networks, which possess a direct link from source to sink. We characterize the optimal DMT of KPP(D) networks with K >= 4, and KPP(I) networks with K >= 3. Along the way, we derive lower bounds for the DMT of triangular channel matrices, which are useful in DMT computation of various protocols. As a special case, the DMT of two-hop relay network without direct link is obtained. Two key implications of the results in the two-part paper are that the half-duplex constraint does not necessarily entail rate loss by a factor of two, as previously believed and that, simple AF protocols are often sufficient to attain the best possible DMT.

Critical lattice size limit for synchronized chaotic state in one- and two-dimensional diffusively coupled map lattices

We consider diffusively coupled map lattices with P neighbors (where P is arbitrary) and study the stability of the synchronized state. We show that there exists a critical lattice size beyond which the synchronized state is unstable. This generalizes earlier results for nearest neighbor coupling. We confirm the analytical results by performing numerical simulations on coupled map lattices with logistic map at each node. The above analysis is also extended to two-dimensional P-neighbor diffusively coupled map lattices.