Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities

In this paper we consider instabilities of localised solutions in planar neural field firing rate models of Wilson-Cowan or Amari type. Importantly we show that angular perturbations can destabilise spatially localised solutions. For a scalar model with Heaviside firing rate function we calculate symmetric one-bump and ring solutions explicitly and use an Evans function approach to predict the point of instability and the shapes of the dominant growing modes. Our predictions are shown to be in excellent agreement with direct numerical simulations. Moreover, beyond the instability our simulations demonstrate the emergence of multi-bump and labyrinthine patterns. With the addition of spike-frequency adaptation, numerical simulations of the resulting vector model show that it is possible for structures without rotational symmetry, and in particular multi-bumps, to undergo an instability to a rotating wave. We use a general argument, valid for smooth firing rate functions, to establish the conditions necessary to generate such a rotational instability. Numerical continuation of the rotating wave is used to quantify the emergent angular velocity as a bifurcation parameter is varied. Wave stability is found via the numerical evaluation of an associated eigenvalue problem

Expression of basic fibroblast growth factor, FGFR1 and FGFR2 in normal and malignant human breast, and comparison with other normal tissues.

The expression of basic fibroblast growth factor (bFGF) and two of its receptors, FGFR1 and FGFR2, was detected using the polymerase chain reaction, and quantified by comparison to the relative amount of product obtained following co-amplification of the ubiquitous glyceraldehyde phosphate dehydrogenase transcript. Varying levels were found in the vast majority of both cancer and non-malignant breast biopsies as well as in samples of several other normal human tissues. Significantly less bFGF was present in cancers (P less than 0.0001). Similarly, FGFR2 product was also much less in cancer tissues (P = 0.0078), as was FGFR1 (P = 0.002). FGFR1 levels in cancers tended to be higher in those which were oestrogen receptor positive (P less than 0.06). Amplification of different coding regions showed evidence of variant forms of FGFR1 RNA. Cancers appeared to have a significantly greater proportion of PCR product corresponding to the region between the third immunoglobulin like domain and the tyrosine kinase domain (P = 0.046). Differential expression was observed in breast cell lines, with bFGF in the normal derived HBL100, HBR SV1.6.1 and 184A1 but little or none in ZR-75-1, MCF-7, T47D and MDA-MB-231. FGFR1 was present in most of these but FGFR2 was absent from T47D, MDA-MB-231 and HBL100. ZR-75-1 cells had a marked preponderance of FGFR1 variants lacking part of the coding sequence. Aberrant receptor processing may provide clues concerning the role of FGF's and their potential involvement in malignancy

Transcripts for transforming growth factors in human breast cancer: clinical correlates.

The levels of mRNA for transforming growth factors (TGF alpha and beta) and the epidermal growth factor receptor (EGFR) were determined in 69 human breast carcinomas and 20 biopsies of non-neoplastic breast tissue by dot blot hybridisation analysis. TGF alpha mRNA was detected in 42% of cancers and 44% of non-neoplastic breast tissue at low levels. TGF beta mRNA was found in all breast cancers and non-neoplastic breast tissues, but the levels of TGF beta mRNA were found to be higher in breast cancers (P = 0.01). EGFR mRNA was detected in 55% of breast cancers and in all non-neoplastic breast tissue tested. The presence of EGFR mRNA was inversely related to oestrogen receptor (ER) status (P = 0.0001). Coexpression of TGF alpha and EGFR was observed in 28% of the carcinomas, and significantly more commonly in ER negative tumours (P = 0.01). No significant relationship was found between histological grade, tumour cellularity or tumour desmoplasia and expression of either the TGFs or of EGFR mRNA. High levels of TGF beta were, however, associated with the absence of lymph node metastases at presentation (P = 0.05). Levels of TGF alpha and beta and EGFR mRNA were analysed in relationship to the relapse-free and overall survival of patients with breast cancer, but none was found to predict significantly the outcome in these patients. Longer clinical follow-up and larger numbers of patients are required to determine whether TGFs will prove a useful marker for prognosis in breast cancer patients