<PAGE>   1
================================================================================

                       SECURITIES AND EXCHANGE COMMISSION
                             WASHINGTON, D. C. 20549

                             ----------------------

                                    FORM 10-Q

              QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d) OF
                       THE SECURITIES EXCHANGE ACT OF 1934

                             ----------------------

                  FOR THE QUARTERLY PERIOD ENDED MARCH 31, 1998

                         COMMISSION FILE NUMBER 0-19874

                             ----------------------

                               NEUREX CORPORATION
             (Exact name of registrant as specified in its charter)

                DELAWARE                                  77-0128552
    (State or other jurisdiction of                    (I.R.S. Employer
     incorporation or organization)                  Identification No.)

              3760 HAVEN AVENUE, MENLO PARK, CALIFORNIA 94025-1012
                    (Address of principal executive offices)

                                 (650) 853-1500

              (Registrant's telephone number, including area code)

                             ----------------------

           SECURITIES REGISTERED PURSUANT TO SECTION 12(B) OF THE ACT:

     Title of each class                 Name of each exchange on which
COMMON STOCK, $ .01 PAR VALUE        registered NASDAQ NATIONAL MARKET SYSTEM

           SECURITIES REGISTERED PURSUANT TO SECTION 12(g) OF THE ACT:

                                      NONE

        INDICATE BY CHECK MARK WHETHER THE REGISTRANT (1) HAS FILED ALL REPORTS
REQUIRED TO BE FILED BY SECTION 13 OR 15(d) OF THE SECURITIES EXCHANGE ACT OF
1934 DURING THE PRECEDING 12 MONTHS (OR FOR SUCH SHORTER PERIOD THAT THE
REGISTRANT WAS REQUIRED TO FILE SUCH REPORTS), AND (2) HAS BEEN SUBJECT TO
FILING REQUIREMENTS FOR THE PAST 90 DAYS. YES [X]  NO [ ]

        THE NUMBER OF SHARES OF COMMON STOCK OUTSTANDING AT APRIL 30, 1998 WAS
22,508,348 SHARES.

================================================================================





                                       1
<PAGE>   2

This report on Form 10-Q contains 14 pages.
================================================================================




                               NEUREX CORPORATION

                                      INDEX


<TABLE>
<CAPTION>
                                                                                       PAGE
 ITEM                           PART I. FINANCIAL INFORMATION                         NUMBER
                                                                                      ------
  <S>  <C>                                                                              <C>
  1.   Financial Statements (unaudited):

       a.  Condensed Consolidated Balance Sheets - March 31, 1998 and December 31,
           1997                                                                          3

       b.  Condensed Consolidated Statements of Operations - Three Months Ended
           March 31,   1998 and 1997                                                     4

       c.  Condensed Consolidated Statements of Cash Flows - Three Months Ended
           March 31, 1998 and 1997                                                       5

       d.  Notes to Condensed Consolidated Financial Statements                         6-7


  2.   Management's Discussion and Analysis of Financial Condition and Results of
           Operations                                                                   8-9


                           PART II. OTHER INFORMATION

  1.   Legal Proceedings                                                                10

  2.   Changes in Securities                                                            10

  3.   Defaults Upon Senior Securities                                                  10

  4.   Submission of Matters to a Vote of Security Holders                              10

  5.   Other Information                                                                10

  6.   Exhibits and Reports on Form 8-K                                                 10

       Signatures                                                                       11
</TABLE>






                                       2
<PAGE>   3


                         PART I - FINANCIAL INFORMATION


ITEM 1.    FINANCIAL STATEMENTS

                               NEUREX CORPORATION

                      CONDENSED CONSOLIDATED BALANCE SHEETS
                                   (UNAUDITED)
               (IN THOUSANDS, EXCEPT PER SHARE AND SHARE AMOUNTS)


<TABLE>
<CAPTION>
                                     ASSETS

                                                                     MARCH 31,      DECEMBER 31,
                                                                       1998            1997
                                                                     ---------      ------------
<S>                                                                  <C>             <C>      
Current assets:
  Cash and cash equivalents                                          $   3,324       $  11,297
  Short-term investments                                                43,119          46,121
  Trade receivables, net                                                 1,990              --
  Other current assets                                                     460             277
                                                                     ---------       ---------
    Total current assets                                                48,893          57,695
Property and equipment, net                                              2,808           2,806
Notes receivable from officers                                              88              94
Other assets, net                                                           67              69
                                                                     ---------       ---------
                                                                     $  51,856       $  60,664
                                                                     =========       =========

                      LIABILITIES AND STOCKHOLDERS' EQUITY

Current liabilities:
  Accounts payable                                                   $   2,924       $   1,912
  Accrued wages and benefits                                               632             691
  Accrued clinical and preclinical testing                               5,456           5,374
  Other accrued liabilities                                              1,049             749
  Prepaid milestone repayable to Medtronic, Inc., a stockholder          1,921           1,870
  Deferred revenue                                                       1,000           1,000
  Current portion of capital lease obligations                             159             178
                                                                     ---------       ---------
    Total current liabilities                                           13,141          11,774
Long-term capital lease obligations                                         69             114
Commitments
Stockholders' equity:
  Convertible preferred stock, $ .01 par value; authorized:
    15,000,000 shares; none outstanding                                     --              --
  Common stock, $ .01 par value; authorized: 45,000,000
    shares; issued and outstanding: 22,355,570 shares at
    March 31, 1998 and 22,239,267 shares at
    December 31, 1997                                                      224             222
  Additional paid-in capital                                           156,700         156,299
  Deferred compensation                                                     --              (5)
  Shareholder receivable                                                    (1)            (48)
  Unrealized loss on investments                                           (29)             (1)
  Accumulated deficit                                                 (118,248)       (107,691)
                                                                     ---------       ---------
    Total stockholders' equity                                          38,646          48,776
                                                                     ---------       ---------
                                                                     $  51,856       $  60,664
                                                                     =========       =========
</TABLE>


                            See accompanying notes.





                                       3
<PAGE>   4

                               NEUREX CORPORATION

                 CONDENSED CONSOLIDATED STATEMENTS OF OPERATIONS

                                   (UNAUDITED)
                    (IN THOUSANDS, EXCEPT PER SHARE AMOUNTS)


<TABLE>
<CAPTION>
                                                        THREE MONTHS ENDED
                                                              MARCH 31,
                                                     -------------------------- 
                                                       1998              1997
                                                     --------          -------- 
<S>                                                  <C>               <C>     
Revenues:
  Product sales, net                                 $  2,100          $     --
  Revenues from collaborative
     agreements and grants:
     Related parties                                      292               427
     Other                                                 --                11
                                                     --------          -------- 
          Total revenues                                2,392               438

Costs and expenses:
     Cost of sales                                        234                --
     Research and development                           8,826             7,608
     Selling, general and
        administrative                                  4,637             1,509
                                                     --------          -------- 
          Total costs and expenses                     13,697             9,117
                                                     --------          -------- 
Loss from operations                                  (11,305)           (8,679)
Interest income, net                                      748             1,037
                                                     --------          -------- 
Net loss                                             $(10,557)         $ (7,642)
                                                     ========          ======== 
Basic and diluted net loss per
    share                                            $  (0.47)         $  (0.35)
                                                     ========          ======== 
Shares used in computing
  basic and diluted net loss per
  common share                                         22,337            22,078
</TABLE>


                             See accompanying notes.





                                       4
<PAGE>   5

                               NEUREX CORPORATION

                 CONDENSED CONSOLIDATED STATEMENTS OF CASH FLOWS

                INCREASE (DECREASE) IN CASH AND CASH EQUIVALENTS

                                   (UNAUDITED)
                                 (IN THOUSANDS)


<TABLE>
<CAPTION>
                                                                 THREE MONTHS ENDED
                                                                      MARCH 31,
                                                              ----------------------- 
                                                                1998           1997
                                                              --------       -------- 
<S>                                                           <C>            <C>      
Cash flows used for operating activities:
  Net loss                                                    $(10,557)      $ (7,642)
    Adjustments to reconcile net loss to net cash used
    for operating activities:
        Depreciation and amortization                              247            146
        Noncash expenses from stock, debt and warrant
        issuances                                                   12             38
    Changes in assets and liabilities:
        Current assets                                          (2,167)            16
        Current liabilities                                      1,378           (349)
        Accrued clinical and preclinical testing                    82          2,117
                                                              --------       -------- 
        Net cash used for operating activities                 (11,086)        (5,676)
Cash flows from investing activities:
  Purchase of property and equipment                              (248)          (428)
  Purchases of short-term investments                          (30,861)       (66,344)
  Sales and maturities of short-term investments                33,835         57,481
                                                              --------       -------- 
        Net cash provided by (used for) investing                2,726         (9,290)
        activities
Cash flows from financing activities:
  Sales of common stock                                            451            131
  Payments of capital lease obligations                            (64)           (53)
                                                              --------       -------- 
        Net cash provided by financing activities                  387             78
                                                              --------       -------- 
Net decrease in cash and cash equivalents                       (7,973)       (14,888)
Cash and cash equivalents at beginning of period                11,297         40,552
                                                              --------       -------- 
Cash and cash equivalents at end of period                       3,324         25,664
Short-term investments at end of period                         43,119         55,489
                                                              --------       --------
Cash, cash equivalents and short-term investments at end
  of period                                                   $ 46,443       $ 81,153
                                                              ========       ========
</TABLE>


                             See accompanying notes.





                                       5
<PAGE>   6

                               NEUREX CORPORATION

                         NOTES TO CONDENSED CONSOLIDATED
                              FINANCIAL STATEMENTS


1.      SUMMARY OF SIGNIFICANT ACCOUNTING POLICIES

        Organization

        Neurex was incorporated in Delaware on October 15, 1986 to develop
products for the treatment of diseases based upon advances in neuroscience
technology and other therapeutic areas with unmet medical needs.

        Principles of consolidation

        The consolidated financial statements include the accounts of Neurex and
its wholly-owned subsidiary. All significant intercompany accounts and
transactions have been eliminated.

        Interim financial statements

        The balance sheet at December 31, 1997 has been derived from audited
financial statements at that date. The information at March 31, 1998, and for
the three month periods ended March 31, 1998 and 1997, is unaudited, but in the
Company's opinion, the accompanying condensed interim financial statements
include all adjustments, consisting only of normal recurring adjustments, which
the Company considers necessary to fairly state the Company's financial position
and the results of operations and cash flows. Certain information and footnote
disclosures normally included in financial statements prepared in accordance
with generally accepted accounting principles have been condensed or omitted.
The accompanying condensed financial statements should be read in conjunction
with the financial statements and notes thereto included in the Company's Annual
Report on Form 10-K for the year ended December 31, 1997. The results of the
Company's operations for any interim period are not necessarily indicative of
the results of the Company's operations for any other interim period or for a
full fiscal year.

        Cash and investments

        Cash and cash equivalents consist of cash held in U.S. banks, time
deposits and other highly liquid investments with a maturity of 90 days or less
from the date of purchase. Cash equivalents are readily convertible into cash
and have insignificant interest rate risk. Short-term investments include
corporate fixed income obligations and U.S. government notes with a maturity of
90 days to three years from the date of purchase. The Company's investment
policy stipulates that a diversified portfolio be maintained and invested in a
manner appropriate for the Company's primary business operations. The policy
defines investment objectives to provide optimal investment return within
constraints that optimize safety and liquidity.

        Management determines the appropriate classification of debt securities
at the time of purchase and reevaluates such designation as of each balance
sheet date. All debt securities to date have been classified as
available-for-sale. Available-for-sale securities are carried at fair value,
with the unrealized gains and losses reported in a separate component of
stockholders' equity. The cost of debt securities is adjusted for amortization
of premiums and accretion of discounts to maturity. Such amortization and
accretion is included in interest income and expense. Realized gains and losses
and declines in value judged to be other-than-temporary on available-for-sale
securities are included in interest income or expense. The cost of securities
sold is based on the specific identification method. Interest and dividends on
securities classified as available-for-sale are included in interest income.

        Revenue Recognition





                                       6
<PAGE>   7

        The Company recognizes product sales revenue at the time product is
shipped. Provisions are made for estimated product returns, cash discounts and
government discounts and rebates. Revenues recognized under the Company's
collaborative agreements, license and supply agreements and government grants
are recorded as earned based upon performance requirements of the contract.
Payments received in advance under these agreements are recorded as deferred
revenues until earned.

        Net loss per share

        In 1997, the Financial Accounting Standards Board issued Statement No.
128, "Earnings per Share." Statement 128 replaced the calculation of primary and
fully diluted earnings per share with basic and diluted earnings per share.
Unlike primary earnings per share, basic earnings per share excludes any
dilutive effects of options, warrants and convertible securities. Diluted
earnings per share is very similar to the previously reported fully diluted
earnings per share. Common equivalent shares from stock options and warrants are
excluded from the diluted computation as their effect is antidilutive. All
earnings per share amounts for all periods have been presented, and where
appropriate restated, to conform to the Statement 128 requirements.

        New Accounting Standards

        In June 1997, the Financial Accounting Standards Board issued Statement
of Financial Accounting Standard No. 131, "Disclosures about Segments of an
Enterprise and Related Information" ("SFAS 131"). The Statement is effective for
fiscal years beginning after December 15, 1997. SFAS 131 establishes standards
for reporting financial and descriptive information about an enterprise's
operating segments in its annual financial statements and selected segment
information in interim financial reports. Reclassification or restatement of
comparative financial statements or financial information for earlier periods is
required upon adoption of SFAS 131. Application of the Statements' disclosure
requirements will have no impact on the Company's consolidated financial
position, results of operations or earnings per share data as currently
reported.

        As of January 1, 1998, the Company adopted Statement 130, Reporting
Comprehensive Income. Statement 130 establishes new rules for the reporting and
display of comprehensive income or loss and its components; however, the
adoption of this Statement had no impact on the Company's net loss or
shareholders' equity. Statement 130 requires unrealized gains and losses on the
Company's available-for-sale securities and foreign currency translation
adjustments, which prior to adoption were reported separately in shareholders'
equity to be included in other comprehensive income. Prior year financial
statements have been reclassified to conform to the requirements of Statement
130.

        During the first quarter of 1998 and 1997, the total comprehensive loss
amounted to $10,586,000 and $7,696,000, respectively.


2.      AVAILABLE-FOR-SALE SECURITIES

        The Company's short-term investments are available-for-sale and consist
primarily of U.S. Treasury Notes and other U.S. government securities of
$10,526,000 and corporate debt securities of $32,502,000 at March 31, 1998. The
gross realized gains and losses on sales of available-for-sale securities were
insignificant in the three months ended March 31, 1998.





                                       7
<PAGE>   8

ITEM 2.    MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND
           RESULTS OF OPERATIONS

OVERVIEW

        Since commencement of operations in October 1986, Neurex has devoted the
majority all of its resources to its research and development programs. The
Company has been unprofitable since inception and expects to incur significant
and increasing losses over at least the next few years in order to continue
clinical development and to commercialize its products. As of March 31, 1998,
the Company's cumulative net loss was $118,248,000. The Company's principal
sources of working capital have been public and private equity financings,
convertible notes payable, milestone and other payments from collaborative
research and development agreements, license fees, interest income, lease
financings and research grants.

        The Company's business is subject to significant risks, including, but
not limited to the success of its research, development, commercialization,
product acceptance and capital raising efforts, uncertainties associated with
obtaining and enforcing patents important to the Company's business, the lengthy
and expensive regulatory process, and possible competition from other products.
Even if the Company's products appear promising at an early stage of
development, they may not reach the market for a number of reasons. Such reasons
include, but are not limited to, the possibilities that the potential products
will be found ineffective during clinical trials, fail to receive the necessary
regulatory approvals, be difficult to manufacture on a large scale, be
uneconomical to market or be precluded from commercialization by the proprietary
rights of third parties. Additional expenses, delays and losses of opportunities
that may arise out of these and other risks could have a material adverse impact
on the Company's financial condition, results of operations and cash flows.

        This Report contains forward-looking statements relating to future
expense levels and financial results. These statements involve inherent risks
and uncertainties. The Company's actual results may differ significantly from
the results discussed in the forward-looking statements. Factors that might
cause such a difference include, but are not limited to, the risks described in
this Overview and discussed in the Company's Annual Report on Form 10-K for the
year ended December 31, 1997.


RESULTS OF OPERATIONS

Three Months Ended March 31, 1998 and 1997

        Revenues were $2,392,000 and $438,000 for the three months ended March
31, 1998 and 1997, respectively. Revenues in 1998 consisted primarily of
CORLOPAM(R) product sales totaling $2,100,000 in the United States. The product
sales for the 1998 period reflect the first quarter of CORLOPAM product sales
since the Food & Drug Administration's ("FDA") approval of this drug on
September 23, 1997. Revenues in 1997 consisted primarily of expense
reimbursements from a related party and included no product sales of
CORLOPAM(R).

        Research and Development expenses increased by $1,218,000 or 16%, to
$8,826,000 for the three months ended March 31, 1998, compared to $7,608,000 in
the earlier period. The increase was due primarily to increased Phase III
clinical study expenses of ziconotide for malignant pain and employment related
expenses. The Company expects research and development expenses to increase
significantly over the next several years.

        Selling, general and administrative expenses were $4,637,000 and
$1,509,000 for the three months ended March 31, 1998, and 1997, respectively,
representing an increase of 207%. This increase is attributable primarily to
increased selling and marketing development activities for the January 1998
launch of CORLOPAM(R) in the United States, as well as employment related
expenses to support a field sales force. The Company expects selling, general
and administrative expenses to continue to increase over the next several years.

        Net interest income decreased to $748,000 for the three months ended
March 31, 1998, compared to $1,037,000 in the earlier period. The decrease in
net interest income was due to a decrease in cash





                                       8
<PAGE>   9

available for investments as the result of funding the Company's operating loss.


LIQUIDITY AND CAPITAL RESOURCES

        For the three months ended March 31, 1998, cash expenditures for
operating activities and additions to capital equipment were $11,334,000. The
Company anticipates that these expenditures will increase significantly in
future periods.

        The Company had available cash, cash equivalents and short-term
investments of $46,443,000 at March 31, 1998. Cash in excess of immediate
requirements is invested according to the Company's investment policy, which
provides guidelines with regard to liquidity and return and, wherever possible,
seeks to minimize the potential effects of concentration and degrees of risk.

        The Company expects to continue to incur substantial additional
operating losses from costs related to the CORLOPAM(R) commercial product launch
activities and the continuation and expansion of research and development,
including clinical studies and increased administrative activities over at least
the next few years. The Company anticipates that its existing capital resources
supplemented by net product revenues and interest earned thereon will enable it
to maintain its current and planned operations through the foreseeable future.
However, the Company's requirements may change depending on numerous factors,
including, but not limited to, the research and development programs, the
results of clinical studies, the number and nature of the indications the
Company pursues in clinical studies, the timing of domestic and foreign
regulatory approvals, technological advances, determinations as to the
commercial potential of the Company's products and the status of competitive
products. In addition, expenditures will be dependent on the establishment of
collaborative relationships with other companies, the availability of financing
and other factors. The Company plans to continue to fund its short and long-term
operations using a combination of public and private equity and debt offerings,
payments from the licensing, sublicensing and sales of its intellectual property
rights and sales of CORLOPAM(R). If such funds are not obtained, the Company may
need to delay or curtail some of its research and development activities.

        On April 30, 1998, the Company announced that it had signed a definitive
merger agreement with Elan Corporation, plc ("Elan") pursuant to which a
wholly-owned subsidiary of Elan will merge with and into the Company. The
Company shall survive the merger and become a wholly-owned subsidiary of Elan.
Under the terms of the agreement, each share of the Company's common stock be
will exchanged for 0.51 of an Elan American Depository Share (the "Exchange
Ratio"). Certain outstanding options and rights to purchase the Company's common
stock will be assumed by Elan and will become an option or right to purchase
Elan American Depository Shares, with appropriate adjustments to the number of
shares issuable thereunder and the exercise price thereof based on the Exchange
Ratio. The merger has been approved by the boards of directors of the Company
and Elan, but is still subject to various customary conditions, including,
without limitation, clearance under the Hart-Scott-Rodino Antitrust Improvements
Act, clearance by Irish and other U.S. regulatory authorities, and the approval
by the stockholders of the Company.









                                       9
<PAGE>   10

                           PART II - OTHER INFORMATION

                               NEUREX CORPORATION


1.      LEGAL PROCEEDINGS

               None.

2.      CHANGES IN SECURITIES

               None.

3.      DEFAULTS UPON SENIOR SECURITIES

               None.

4.      SUBMISSION OF MATTERS TO A VOTE OF SECURITY HOLDERS

               None.

5.      OTHER INFORMATION

        On April 29, 1998, the Company executed a definitive merger agreement
        (the "Merger Agreement") with Elan Corporation, plc ("Elan"), a leading
        specialty pharmaceutical company. Pursuant to the Merger Agreement, the
        Company will merge with, and become, a wholly-owned subsidiary of Elan.
        At the effective time of the merger, each outstanding share of the
        Company's common stock will be converted into the right to receive .51
        of an Elan American Depository Share. The respective obligations of the
        Company and Elan to effect the merger are subject to the satisfaction of
        certain conditions, including, but not limited to, obtaining requisite
        stockholder and Irish and U.S. regulatory approvals, the absence of any
        injunction prohibiting consummation of the merger, the accuracy of the
        representations and warranties contained in the Merger Agreement, the
        receipt of certain legal opinions with respect to tax matters and the
        receipt and confirmation of certain accountants' letters with respect to
        the accounting treatment of the transaction.

        On April 30, 1998, the Company filed with the Commission, a Current
        Report on Form 8-K containing additional details of the transaction. The
        Merger Agreement was filed as an exhibit to the Form 8-K.


6.      EXHIBITS AND REPORTS ON FORM 8-K

               (a)  Reports on Form 8-K

                    During the quarter ended March 31, 1998, the Company filed
                    no Current Reports on Form 8-K.






                                       10
<PAGE>   11

                               NEUREX CORPORATION

                                   SIGNATURES



        Pursuant to the requirements of the Securities Exchange Act of 1934, the
registrant has duly caused this report to be signed on its behalf by the
undersigned thereunto duly authorized.



Date:   May 14, 1998                     Neurex Corporation
        --------------




By:     /S/ JOHN VARIAN
        ------------------------------------
        John Varian
        Vice President and Chief
        Financial Officer (Principal
        Financial and Accounting Officer
        and Duly Authorized Officer)







                                       11


<PAGE>   12
                               INDEX TO EXHIBITS
 

Exhibit
Number                   Description
------                   -----------

 27            Financial Data Schedule

 27.1          Restated Financial Data Schedule

 27.2          Restated Financial Data Schedule